Algebra II∗

Introduction to Algebraic Geometry

Alexandru Constantinescu

29th June 2020

Freie Universität BerlinSommersemester 2020

∗ These are not official lecture notes, but simply notes taken while preparing the lectures. Mistakes can and will occur.

Contents

1 Basic Notions 5

2 Closed subsets of the affine space 62.1 The Zariski Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Hilbert’s Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Irreducible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Consequences of Hilbert’s Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Polynomial Maps and The Coordinate Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 The relative version of Z and I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.9 Rational Functions and Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.10 Regular Functions on Quasi-affine Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Closed Subsets of Projective Space 143.1 Quasiprojective Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 The Function Field of an Irreducible Quasi-Projective Variety . . . . . . . . . . . . . . . . . . . . 163.3 Products of Quasi-Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 The Segre Embedding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.4 Fiber Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Linear Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Finite sets of Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 The Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6.1 The exterior product of two vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6.2 The relation with two-dimensional subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 203.6.3 Exterior powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6.4 The Plücker Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6.5 Affine Cover of the Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.7 Regular Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.7.1 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.7.2 Consequences of being closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.7.3 Finite Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.7.4 Noether Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.8 Rational Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Dimension 324.1 Intersections with Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Lower Bounds on Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Dimension Of Fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 Criterion for Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2

5 Singularities 395.1 The Local Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 The Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Intrinsic Definition of the Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.4 Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.4.1 On an irreducible variety X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4.2 On any variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 The Tangent Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.6 Local Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.7 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.8 Properties of Nonsingular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Bibliography

[Website] http://userpage.fu-berlin.de/aconstant/Alg2.html

[AM69] Michael Francis Atiyah, Ian Grant Macdonald, Introduction to Commutative Algebra, Addison-WesleyPublishing Company, 1969

[Art91] Michael Artin, Algebra, Pearson, 1991.

[Mun00] James Munkres, Topology, 2ed., (Mainly Chapter 2) Prenctice Hall, US, 2000.

[Stacks] The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu, 2019.

[Eis95] David Eisenbud, Commutative Algebra: with a View toward Algebraic Geometry, Springer Science &Business Media, 1995.

[Hul03] Klaus Hulek, Elementary Algebraic Geometry, American Mathematical Society, Student MathematicalLibrary, Vol. 20, 2003.

[Har77] Robin Hartshorne, Algebraic Geometry,Graduate Texts in Mathematics, republished 2013, SpringerNew York, 1977.

[Har92] Joe Harris, Algebraic Geometry: A First Course, Graduate Texts in Mathematics, Springer, 1992.

[Gat19] Andreas Gathmann, Algebraic Geometry, Class Notes TU Kaiserslautern 2019/20, https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2019/alggeom-2019.pdf.

[Sha13] Igor R Shafarevich, Basic algebraic geometry; 3rd ed., Springer, Berlin", 2013.

Introduction

“Algebraic geometry is the study of geometries that come from algebra [. . . ]”1Who can argue with that? I will not try to give a definition of algebraic geometry here. At least, not yet. I also

do not recommend looking for the answer before getting acquainted with some of the definitions and problems.Naive instances of algebraic geometry, which you probably have encountered so far, are: solving linear systems

of equations, finding solutions to polynomial equations in one variable xn + an−1xn−1 + · · · + a1x + a0 = 0,

and the classification of (plane) quadrics. The strategy for linear systems was independent of the underlyingfield: just use Gaussian Elimination. When solving polynomial equations, the answer depends more on the fieldwhere we want the solutions to live. Over algebraically closed fields, we know that solutions always exist, andthat, counting with multiplicities, there are always n. In classifying quadrics, the underlying field played animportant role: the classification over R is more complicated than the one over C. Also, this last problem, is ina sense closest to what algebraic geometry is “really” about.

Instead of saying what algebraic geometry is, we will start looking at some of the concepts which have beenpart of its structure the longest: solution sets of systems of polynomial equations in several variables. We willbriefly recall these definitions (which should have been mentioned in any good course in Commutative Algebra),and then move on and look at functions on them and maps between them. We will do all this first in the affinespace, then in the projective space. The whole subject evolved around some core “examples”: Veronese andSegre embeddings, Grassmannians. We will spend some time looking at them in detail. Much later, we willsee how these objects can we looked at without choosing an embedding. But what would “solution sets” meanwithout an ambient space? Hang on tight, work hard, and you will find out.

1Wolfram Math World

4

Chapter 1

Basic Notions

“Let K be an algebraically closed field.”Albert Einstein

Throughout these notes, unless specifically stated otherwise, K will denote an algebraically closed field. Wewrite K for the algebraic closure, so K = K. In particular, K is an infinite field. We will not make any assump-tions on the characteristic of K.

For n ∈ N we will write AnK, or simply An, for the n-dimensional affine space over K. Its elements are calledpoints and are of the form P = (a1, . . . , an), with ai ∈ K. So An is identified as a set with Kn, when n ≥ 1.While it may not come up here, it is good to know that, for n = 0, we have that A0 consist of one point.

Rings are all considered to be commutative and have a 1. Ring homomorphisms are required to map 1 to 1.The radical of an ideal I of a ring R is

√I = {f ∈ R | ∃ r > 0 such that fr ∈ I}. An ideal is radical if I =

√I .

For any subset T ⊆ R we will write 〈T 〉 for the ideal generated by T .We will stick as much as possible to the notation oh Hartshorne’s book [Har77]. For instance, we will use A todenote the polynomial ring K[x1, . . . , xn] in n variables with coefficients in K. Every polynomial is a K−linearcombination of monomials, so a finite sum of the form

f =∑ci1,...,inx

i11 · · ·xinn ,

with ci1,...,in ∈ K. The degree of a monomial xi11 · · ·xinn is the sum of the exponents: i1 + · · ·+ in. The degreeof a polynomial is the maximum degree of a monomial appearing in f with nonzero coefficient. Recall theuniversal property of the polynomial ring: for any K−algebra R, and any a1, . . . , an ∈ R, there exists a uniquering homomorphism, called the evaluation map eva : K[x1, . . . , xn] −→ R with xi 7−→ ai. We will denote byf(a1, . . . , an) := eva(f) ∈ R. This allows us to interpret polynomials f ∈ A as functions f : An −→ K in theobvious way.

5

Chapter 2

Closed subsets of the affine space

2.1 The Zariski Topology

Definition 2.1. Let T ⊆ A = K[x1, . . . , xn] be an arbitrary set of polynomials. The zero set of T is

Z(T ) := {P ∈ An | f(P ) = 0 ∀ f ∈ T}.

A subset X ⊆ An is called an affine algebraic set (or just algebraic set) if there exists T ⊆ A such thatX = Z(T ).

Remark 2.2. (i) Some authors call affine algebraic sets affine varieties (e.g. [Har92, Gat19]), or refer tothem only as closed sets. Shafarevich calls affine varieties sets which are isomorphic to affine algebraic sets[Sha13] (which we will see to be more general). Hartshorne [Har77] uses affine varieties only for irreducibleaffine algebraic sets. We will try to be as specific as possible with our use of the term, but caution andcommon sense are advised.

(ii) It is easy to check that Z(T ) = Z(〈T 〉): if f, g ∈ T and h ∈ A, then Z(T ) = Z(T ∪{f+g}) = Z(T ∪{hf}).By Hilbert’s Basis Theorem, the polynomial ring A is Noetherian, so there exists a finite set of generatorsf1, . . . , fr of 〈T 〉. In particular, every affine algebraic set is the zero set of an ideal, and the zero set of afinite number of polynomials. In the latter case, we just write Z(f1, . . . , fr) for Z({f1, . . . , fr}).

Proposition 2.3. For any T1, T2 ⊆ A and any family {Ti}i∈I of subsets of A we have

1. If T1 ⊆ T2, then Z(T1) ⊇ Z(T2).

2. Z(T1) ∪ Z(T2) = Z(T1T2), where T1T2 = {f1 · f2 | fi ∈ Ti}.

3.⋂i∈I Z(Ti) = Z

(⋃i∈I Ti

).

4. ∅ = Z(1).

5. An = Z(0).

Proof. Very easy. Check [Har77, Proposition 1.1]

By Proposition 2.3 we can introduce the following.

Definition 2.4. The Zariski Topology on An is defined by taking the affine algebraic sets as closed sets.

So the open sets in the Zariski Topology are complements of zero sets. Definition 2.4 can be defined for anyfield K, not only for algebraically closed fields. When there is some metric on Kn, the Zariski Topology is verydifferent from the usual topology induced by the metric. This is another reason we write An instead of Kn,even if the two coincide as sets.

6

Examples. 1. Besides A1 and ∅, the algebraic sets in A1 are finite collections of points.

2. A1 \ {0} is not algebraic. (This holds only if the field is infinite, which is always the case for algebraicallyclosed fields).

3. If X ⊆ An and Y ⊆ Am are algebraic, then X × Y ⊆ An × Am is algebraic. If X = Z(f1, . . . , fr), withfi ∈ K[x1, . . . , xn], and Y = Z(g1, . . . , gs), with gi ∈ K[y1, . . . , ym], then X × Y = Z(f1, . . . , fr, g1, . . . , gs)with fi, gj ∈ K[x1, . . . , xn, y1, . . . , ym].

2.2 Hilbert’s Nullstellensatz

Definition 2.5. For any subset Y ⊆ An we define the ideal of Y in A = K[x1, . . . , xn] as

I(Y ) := {f ∈ A : f(P ) = 0 ∀ P ∈ Y }.

It is easy to check that this is actually an ideal of the polynomial ring.

Proposition 2.6. For any Y, Y1, Y2 ⊆ An we have:

(i) I(Y1 ∪ Y2) = I(Y1) ∩ I(Y2).

(ii) I(Y1 ∩ Y2) =√I(Y1) + I(Y2) .

(iii) If Y1 ⊆ Y2, then I(Y1) ⊇ I(Y2).

(iv) Z(I(Y )) = Y , the closure of Y .

Proof. See Hartshorne Proposition 1.2., but it is mostly very straight forward. Point (ii) is stated here a bitearly, but one should be able to prove it easily using the Nullstellensatz.

So now we have the most basic and straight-forward connection between geometry and algebra: a way to goback and forth between the subsets of the affine plane and the subsets of the polynomial ring:

{subsets of K[x1, . . . , xn]} {subsets of AnK}

Z

I

with Z landing in affine algebraic sets, and I landing in radical ideals. As a consequence of the Nullstellensatz weget, what some people call the Nullstellensatz. Notice that under this name there are many different phrasingscirculating. Let us first state the coolest version of this theorem.

Theorem 2.7 (Hilbert’s cool Nullstellensatz). Let K be field and F be a finitely generated K-algebra. If F isa field, then F |K is a finite field extension. That is, F is a finite dimensional K-vector space, thus F is alsoalgebraic over K. In particular, this means that if K is algebraically closed, then F = K.

And here is the most comprehensive, version of the standard Nullstellensatz.

Theorem 2.8 (Hilbert’s Nullstellensatz). Let K be an algebraically closed field and A = K[x1, . . . , xn].

(i) For every maximal ideal m ⊆ A, there exists some point P = (a1, . . . , an) ∈ AnK such that

m = I(P ) = (x1 − a1, . . . , xn − an).

(ii) If I ( A is a proper ideal, then Z(I) 6= ∅.

(iii) For any I ⊆ A we have I(Z(I)) =√I .

Before stating the obvious consequence of the Nullstellensatz, let us talk about irreducible algebraic sets.

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2.3 Irreducible sets

Definition 2.9. Let X be a topological space. A nonempty subset ∅ 6= Y ⊆ X is called irreducible if for anyY1, Y2 closed subsets of Y (that is closed in the induced subspace topology on Y ) we have:

Y = Y1 ∪ Y2 ⇒ Y = Y1 or Y = Y2.

In other words, Y cannot be decomposed as a union of proper closed subsets. A set is reducible if it is notirreducible. It is convenient to say that the empty set is reducible.

Lemma 2.10. For any topological space X and any ∅ 6= Y ⊆ X the following are equivalent:

(i) Y is irreducible.

(ii) For any two nonempty open sets U1, U2 of Y we have U1 ∩ U2 6= ∅.

(iii) Every nonempty open set U ⊆ Y is dense in Y .

Proof. Very easy.

Proposition 2.11. Let X ⊆ An be an algebraic set. We have

X is irreducible ⇔ I(X) is prime.

Proof. Hartshorne Corollary 1.4.

Definition 2.12. A topological space is called Noetherian if every descending sequence of closed sets becomesstationary. Equivalently, if every ascending sequence of open sets becomes stationary.

Remark 2.13. Given the order-reversing correspondence between algebraic sets and ideals of the polynomialring, which is Noetherian by the Hilbert Basis Theorem, the affine space An with the Zariski topology isa Noetherian topological space. So from the Axiom of Choice it follows that every nontrivial collection ofalgebraic sets has a minimal element.

Proposition 2.14. Let X be a Noetherian topological space and ∅ 6= Y ⊆ X be a closed set.

(i) There exist finitely many irreducible closed sets Y1, . . . , Yr such that Y = Y1 ∪ · · · ∪ Yr.

(ii) If we assume the decomposition above to be irredundant, that is if for any i 6= j we have Yi 6⊆ Yj, then thesets Y1, . . . , Yr are uniquely determined.

Definition 2.15. The unique irreducible closed sets Y1, . . . , Yr from Proposition 2.14 are called the irreduciblecomponents of Y .

Proof of Proposition 2.14. See Hartshorne Proposition 1.5, or the video.

2.4 Consequences of Hilbert’s Nullstellensatz

The assumption that K is an algebraically closed field is essential for the following.

Corollary 2.16. The correspondence Z and I between subsets of An and A, induce the following bijections:

{radical ideals of A} 1:1←→ {algebraic sets of An}

⊆ ⊆

{prime ideals of A} 1:1←→ {irreducible algebraic sets of An}

⊆ ⊆

{maximal ideals of A} 1:1←→ {points of An}.

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2.5 Polynomial Maps and The Coordinate Ring

The rough idea behind this section is the following: once we fix a family of geometric objects, here affinealgebraic sets, we want to study what type of functions should be considered, such that the set of all functionson the given set recovers the geometric object. The answer for affine algebraic sets will be polynomial functions.

Definition 2.17. Let X ⊆ An be an affine algebraic set. A polynomial function on X is a map F : X −→ Ksuch that there exists a polynomial f ∈ A with F (P ) = f(P ) for all P ∈ X.

Remark 2.18. The polynomial f is (usually, i.e. if X 6= An) not uniquely determined by the values it takeson X. In particular, for two polynomials f, g ∈ A we have

f(P ) = g(P ), ∀ P ∈ X ⇐⇒ (f − g)(P ) = 0, ∀ P ∈ X ⇐⇒ f − g ∈ I(X).

This remark leads us to the following definition.

Definition 2.19. The affine coordinate ring of an affine algebraic set X ⊆ An is

A(X) := A/I(X).

From Remark 2.18 we get

A(X) = {F : X −→ K : F is a polynomial function}.

Note that the classes of the coordinate functions xi : X −→ K generate A(X) as a K-algebra, hence the namecoordinate ring.

Examples. 1. If P is a point, then A(P ) = K.

2. If X = An, then A(X) = K[x1, . . . , xn].

3. If X = Z(xy − 1) then A(X) = K[x, y]/(xy − 1) ' K[x, x−1] = {f(x)/xn : f(x) ∈ K[x] and n ≥ 0}.

4. For X and Y algebraic sets we have A(X × Y ) ' A(X)⊗K A(Y ).

2.6 The relative version of Z and IThe coordinate ring of An is thus the polynomial ring A = K[x1, . . . , xn]. The Nullstellensatz gives thus acorrespondence between closed subsets of An and ideals of its coordinate ring A. This can be extended to closedsubsets of any algebraic set.

Let X ⊆ An be a fixed affine algebraic set. For any set T ⊆ A(X) of polynomial functions on X, we define the(relative) zero set of T as

ZX(T ) := {P ∈ X : F (P ) = 0, ∀ F ∈ T}.

Subsets of this form are called 1 algebraic subsets of X. For a subset Y ⊆ X, the (relative) ideal of Y inX is defined as

IX(Y ) := {f ∈ A(X) : F (P ) = 0, ∀ P ∈ Y }.

It is again straight-forward to check that Ix(Y ) is an ideal of A(X).

Remark 2.20. An easy check shows that, for an algebraic subset Y of X we have A(Y ) = A(X)/IX(Y ).1Some authors call algebraic sets varieties, and these subsets subvarieties.

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Remark 2.21 (The Relative Nullstellensatz). Just as in Proposition 2.6 (iv) we have for an algebraic set Xand an algebraic subset Y ⊆ X that

ZX(IX(Y )) = Y.

As in Theorem 2.8 (iii) we have for every ideal J ⊆ A(X) that

IX(ZX(I)) =√I .

Together with the description of ideals in a quotient ring, this gives us thus the analogue correspondences ofCorollary 2.16:{

radical ideals J ⊆ Awith I(X) ⊆ J

}1:1←→ {radical ideals of A(X)} 1:1←→ {algebraic subsets of X}

⊆ ⊆ ⊆{prime ideals J ⊆ A

with I(X) ⊆ J

}1:1←→ {prime ideals of A(X)} 1:1←→ {irreducible algebraic sets of X}

⊆ ⊆ ⊆{maximal ideals J ⊆ A

with I(X) ⊆ J

}1:1←→ {maximal ideals of A(X)} 1:1←→ {points of X}.

This also shows that taking on X the topology induced by the Zariski topology on An is the same as taking thetopology given by taking as closed sets those of the form ZX(I).

Note that the coordinate ring of an algebraic setX is not only a ring, but a K-algebra. That is, there exists a ringhomomorphism K −→ A(X). As the kernel must be an ideal and as 1 7→ 1, this homomorphism must always beinjective. This gives A(X) a K-vector space structure, with the ring multiplication map A(X)×A(X) −→ A(X)being K-bilinear. We will always take this extra structure in consideration when dealing with A(X). Althoughcoordinate K-algebra would be more appropriate, the term coordinate ring is too established in the literature.

Recall that every finitely generated K-algebra R is of the form K[x1, . . . , xn]/I, for some n and some ideal Iof the polynomial ring. This comes from the following morphism: K[x1, . . . , xn] −→ R = K[g1, . . . , gn], whereg1, . . . , gn ∈ R are K-algebra generators. Recall also that a ring is reduced if it has no nilpotent elements, anda finitely generated K-algebra is reduced if and only if it is isomorphic to the quotient of the polynomial ringby a radical ideal. This discussion brings us to the following.

Remark 2.22. There is a general 1-to-1 correspondence between affine algebraic sets and reduced K-algebras.Furthermore, from Corollary 2.16 we can deduce the following:

X is irreducible ⇐⇒ A(X) is an integral domain,

So either if we care about affine algebraic sets, or just irreducible affine algebraic sets, there is a correspondencewith some sort of finitely generated K-algebras.

Example 2.23. The parabola X1 = Z(y − x2) has K[X1] ' K[x]. The semicubical parabola X2 = Z(y2 − x3)has K[X2] ' K[x, y]/(y2 − x3) which is not a UFD. This shows there is some difference between the two, evenif there exist bijections A1 −→ X1 given by t −→ (t, t2), and A1 −→ X2 given by t −→ (t2, t3).

2.7 Morphisms

We want now to consider maps between algebraic sets ϕ : X −→ Y with X ⊆ An and Y ⊆ Am. We denote thecoordinate functions on An by x1, . . . , xn and those on Am by y1, . . . , ym. What should these morphisms be?Here is the answer:

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Definition 2.24. A map ϕ : X −→ Y is called a polynomial map if there exists polynomials F1, . . . , Fm ∈K[x1, . . . , xn] such that

ϕ(P ) = (F1(P ), . . . , Fm(P )) ∈ Y ⊆ Am, ∀ P ∈ X.

The next lemma shows that any polynomial map is given by polynomial functions on X.

Lemma 2.25. A map ϕ : X −→ Y is a polynomial map if and only if yj ◦ ϕ ∈ A(X) for j = 1, . . . ,m.

Proof. [Hul03] Lemma 1.41.

To claim that we defined the right concept of morphism, the following is necessary.

Lemma 2.26. Any polynomial map ϕ : X −→ Y is continuous in the Zariski topology.

Proof. Let W ⊆ Y be closed, for instance W = Z(h1, . . . , hr) with hi ∈ A(Y ). Then we have ϕ−1(W ) =Z(h1 ◦ ϕ, . . . , hr ◦ ϕ), thus it is closed.

Remark 2.27. The composition of polynomial maps is a polynomial map. This is very easy to see.

2.8 Pullbacks

The pullback of a function can be defined in the most general setting: for maps between sets.

Definition 2.28. For an arbitrary map between sets ϕ : X −→ Y we can associate to every function on Y , sayf : Y −→ K, a function on X using ϕ by composing ϕ with f :

X Y

K

ϕ

ϕ∗(f) f

The function ϕ∗(f) := f ◦ ϕ is called the pullback of f via ϕ.

If the map ϕ : X −→ Y is a polynomial map between algebraic sets and if f is a polynomial function onY , then the pullback ϕ∗(f) is also a polynomial function on X. Thus every polynomial map induces a mapϕ∗ : A(Y ) −→ A(X). It is very easy to check that this is a ring homomorphism. Also, because ϕ∗(c) = c for aconstant function c, we have ϕ∗ is even a K-algebra homomorphism. For a further map ψ : Y −→ Z, we clearlyhave

(ψ ◦ ϕ)∗ = ϕ∗ ◦ ψ∗.

There is an inverse to the pullback.

Proposition 2.29. If ψ : A(Y ) −→ A(X) is a K-algebra homomorphism, then there exists a unique polynomialmap ϕ : X −→ Y such that ψ = ϕ∗.

Proof. [Hul03, Proposition 1.45]

We have a contravariant functor between the category of affine algebraic sets with polynomial maps as morph-isms, and finitely generated reduced K-algebras with K-algebra homomorphisms.{

algebraic sets,polynomial maps

}−→

{f.g. reduced K-algebras,

K-algebra homomorphisms

}X 7−→ A(X)

ϕ : X −→ Y 7−→ ϕ∗ : A(X) −→ A(Y )

11

This is actually an equivalence of categories (by taking the opposite category for K-algebras). Furthermore, wealso have an equivalence of categories between{

category of irreducibleaffine algebraic sets

}←→

{category of f.g.K-algebras

which are domains

}op.

Definition 2.30. A polynomial map ϕ : X −→ Y is an isomorphism if there is a polynomial map ψ : Y −→ Xsuch that ϕ ◦ ψ = idY and ψ ◦ ϕ = idX .

So in particular, a map ϕ : X −→ Y is an isomorphism of algebraic sets if and only if ϕ∗ : A(Y ) −→ A(X) isan isomorphism of K-algebras.

Example 2.31. The parabola and the cubic parabola are not isomorphic. The first is isomorphic to A1, theother is not, because the inverse map is not polynomial. (cf. [Hul03, Examples 1.50, 1.51]).

2.9 Rational Functions and Maps

Notice that in Hartshorne’s book, affine varieties refer to irreducible affine closed sets. Unfortunately thisconvention is not widely spread, and we will not use it. We will use affine variety in the way Shafarevich does,namely to refer to a quasi-projective variety which is isomorphic to an affine closed set.One reason to focus on irreducible algebraic sets is because each algebraic set is uniquely decomposable as aunion of irreducible components (cf. Proposition 2.14). So maybe we can make things easier like this. Oneadvantage of assuming irreducibility is that the coordinate ring is a domain. This means it has a field offractions.

Definition 2.32. The function filed or field of rational functions of an irreducible affine set X is the fieldof fractions of the integral domain A(X). We denote this by K(X).

Definition 2.33. A rational function ϕ ∈ K(X) is regular at P ∈ X if it can we written as ϕ = f/g, withf, g ∈ A(X) and g(P ) 6= 0. The element f(P )/g(P ) ∈ K is called the value of ϕ at P .

Theorem 2.34. A rational function that is regular at all points of an affine closed set X, is a polynomialfunction on X.

• The set of points at which a rational function ϕ is regular is nonempty and open. It is called the domainof definition of ϕ. As X is irreducible, every nonempty open subset is dense.

• For ϕ1, . . . , ϕm, the set of points at which they are all (“at the same time”) regular is also nonempty anddense.

• A rational function is uniquely determined if by its values on some nonempty open subset.

Consider now ϕ1, . . . , ϕm ∈ K(X). These define a map ϕ : X −→ Am by sending P 7−→ (ϕ1(P ), . . . , ϕm(P )).Thus it is only defined on an open subset U ⊆ X. This type of maps, which are not defined at every point, setsalgebraic geometry apart from topology, or differential geometry.

Definition 2.35. A rational map ϕ : X −→ Y ⊆ Am is an m-tuple of rational functions ϕ1, . . . , ϕm ∈ K(X)such that for all P at which all ϕi are regular we have ϕ(P ) := (ϕ1(P ), . . . , ϕm(P )) ∈ Y . We say that ϕ isregular at such a point. The image of ϕ is

ϕ(X) = {ϕ(P ) : P ∈ X and ϕ is regular at P}.

So for ϕ(X) ⊆ Y we need that (ϕ1, . . . , ϕm), as elements of K(X), satisfy all the equations of Y . I.e., iff(y1, . . . , ym) ∈ IY , then f(ϕ1, . . . , ϕm) = 0 ∈ A(X). So, if ϕ is regular at P , then f(ϕ(P )) = 0, ∀ f ∈ IY .Conversely also: f(ϕ1, . . . , ϕm) ∈ K(X) vanishes on a nonempty open subset of X (the domain of definition),

12

and is thus the zero function on X.

If ϕ(X) is dense in Y , then consider ϕ : U −→ ϕ(X) ⊆ Y , and construct ϕ∗ : A(Y ) −→ K(X). This is aninjective ring homomorphism (Shaf, p.38), which extends to an injective homomorphism ϕ∗ : K(Y ) −→ K(X)

If ϕ(X) is dense in Y and ψ : Y −→ Z is another rational map, then we can compose the two.If ψ(Y ) is dense in Z, then (ψ ◦ ϕ)∗ = ϕ∗ ◦ ψ∗.

Definition 2.36. A map ϕ : X −→ Y is birational if it has a rational inverse (wherever the compositions aredefined). In this case we say that X and Y are birational or birational equivalent.

Closed sets are birational if and only if K(X) ' K(Y ) as fields.Examples (Shaf, p.30, 8-9) hyperbola and cubic parabola.An affine closed set which is birational to an affine space is called affine rational variety.Examples: Quadrics, Cubics with skew lines.

Theorem 2.37. Any irreducible closed set X is birational to a hypersurface of some affine space Am.

Uses: (Shaf P.280)

Proposition 2.38. Let K = K be an algebraically closed field, and K ⊂ L be a finitely generated field extension.Then there exist z1, . . . , zd, zd+1 ∈ L with L = K(z1, . . . , zd+1), with z1, . . . , zd algebraically independent, andzd+1 separable over K(z1, . . . , zd).

(Separable: the derivative of the minimal polynomial is not zero. Always holds in characteristic zero.)

2.10 Regular Functions on Quasi-affine Varieties

Let X be an irreducible affine set. A quasi-affine variety Y is an open subset Y ⊆ X of an irreducible closedaffine set X ⊆ An. The local ring of X at a point P ∈ X is the ring

OX,P := {f ∈ K(X) : f is regular at P}.

This is indeed a local ring, with unique maximal ideal

MP :={f/g ∈ K(X) : f, g ∈ A(X), f(P ) = 0, g(P ) 6= 0

}.

It is the localization of A(X) at the maximal ideal IX(P ). The ring of regular functions on a quasi-affine varietyY ⊆ X is

O(Y ) := OX(Y ) := {f ∈ K(X) : f is regular on Y }.

By convention we set O(∅) := {0}. As rational functions are determined by the values they take on an openset, we have that K(X) = K(Y ).

13

Chapter 3

Closed Subsets of Projective Space

Let V be a finite dimensional K-vector space.

Recall projective planes and the projective space associated to a vector space. (Shaf p.41, Hulek p.66: Lemmafor subspaces)Give the motivation of Bezouts Theorem, and discuss circles in the plane.Recall graded rings and homogeneous elements (also called forms). (Hart. p.9)Notice the difference in exposure between Hart p9 and Shaf p41.

Definition 3.1. A subset X ⊆ Pn is a projective algebraic set (or closed projective set) if it is the set ofpoints at which a collection of polynomials vanishes1. We still write X = Z(T ), with T ⊆ K[x0, . . . , xn].

The Zariski topology is defined analogously with the affine case. So we may apply the same for irreducibilityand dimension.

First define I(X) for an algebraic set, and then prove it must be homogeneous (following Shaf.p41).

We write, following Hartshorne, S(X) = K[x0, . . . , xn]/I(X) for the homogeneous coordinate ring.

Most things hold, with slight modifications. Important difference: the unique homogeneous maximal idealm = (x0, . . . , xn), is called irrelevant because Z(m) = ∅. So the correspondence with points and homogeneousmaximal ideals fails. There is still a 1-1 correspondence.[See Hart p11 ex 2.1-2.7][Hart Lemma on P.44]

Affine pieces of projective space. Hartshorne Proposition 2.2.

Projective completion via homogenization (Again, Gröbner bases come up here when one wants to be practical).Example of homogenizing equations. A non-obvious one?

Cone over a projective variety. (Not done in lecture.)

We will have a discussion about dimension at some point.

1We do not ask explicitly for the polynomials to be homogeneous. However, to for a polynomial to vanish at a point, independentlyof the choice of homogeneous coordinates, all the homogeneous components must vanish.

14

3.1 Quasiprojective Varieties

Definition 3.2. A quasi-projective variety is an open subset of a closed projective set.

Equivalently, a quasi-projective variety is a locally closed2 subset of Pn. Obviously closed projective sets arequasi-projective. We have seen in Exercise 1 on Sheet 3 that if Y is a closed subset of affine space An, which weidentify with one of the affine pieces U0 of Pn, then Y = U0∩Y , where Y is the projective closure (or projectivecompletion) of Y . So affine varieties are also quasi-projective. An example of a quasi-projective variety whichis neither affine nor projective is A2 \ {(0, 0)}.The Zariski topology on a quasi-projective variety is the subspace topology inherited from Pn. A subvarietyof a quasi-projective variety X ⊆ Pn is a subset Y ⊆ X which is itself a quasi-projective variety in Pn. This isequivalent to saying that Y = Z \ Z1 with Z,Z1 ⊆ X closed subsets.Here is one more reason for projective geometry.[Shafarevich:]A quasi-projective variety that is isomorphic to a closed set of affine space will be called affinevariety. It can happen that X itself is not a closed set in An (e.g. X = A1 \ {0}.) So being a closed affineset is not invariant under isomorphism, while being an affine variety is by definition. In the same way, aquasi-projective variety isomorphic to a closed projective set will be called projective variety. We will provein Theorem 1.10 that if X ⊆ Pn is a projective variety, then it is closed in Pn, so that the notions of closedprojective set and projective variety coincide and are both invariant under isomorphism.In general, a property is called local property if it need only be verified for some neighbourhood U of everypoint P ∈ X. In other words, if X =

⋃Uα, with Uα open, it is enough to verify the property on the Uα.

Lemma 3.3. The property that a subset Y ⊆ X is closed in a quasi-projective variety X is a local property.

Proof. We have to show that if X =⋃Uα, and Y ∩ Uα is closed in each Uα, then Y is closed in X.

As the Uα are open, we have Uα = X\Zα, with Zα closed. Also by definition, Uα∩Y is closed if Uα∩Y = Uα∩Tαfor some Tα ⊆ X closed. To conclude, we will show that

Y =⋂(

Zα ∪ Tα).

⊆ Let y ∈ Y . If y ∈ Uα, then y ∈ Y ∩ Uα ⊆ Tα. If y /∈ Uα, then y ∈ X \ Uα = Zα. So Y ⊆⋂

(Zα ∪ Tα).

⊇ Conversely, let x ∈ Zα ∪ Tα for all α. Since X =⋃Uα, then x ∈ Uβ for some β. But this means x /∈ Zβ ,

thus it must be in Tβ , so x ∈ Tβ ∩ Uβ ⊆ Y .

The following lemma shows that quasi-projective varieties are locally affine, in the sense that every point has aneighborhood which is isomorphic to an affine closed set.

Lemma 3.4. Every point x ∈ X of a quasi-projective variety has a neighbourhood which is isomorphic to anaffine variety.

Proof. Shaf. p49. Lemma 1.3. (+ use Lemma 3.6 at the end instead).

Definition 3.5. Let f be a regular function on the affine variety X. An open set D(f) := X \ Z(f) is calledprincipal open set.

Lemma 3.6. Principal open sets are affine varieties.

Proof. Let g1, . . . , gm ∈ K[x1, . . . , xn] be the defining equations of X ⊆ An. Then

D(f) ' Z := Z(g1, . . . , gm, xn+1 · f − 1) ⊆ An+1.

2i.e. the intersection of an open set with a closed set, or, equivalently, if it is an open subset of its closure.

15

The isomorphism is given by ϕ : D(f) −→ Z with

ϕ = (x1, . . . , xn,1

f)

ϕ−1 = (x1, . . . , xn).

Since f(P ) 6= 0 for all P ∈ D(f), the rational function ϕ is regular at every point, thus, by Theorem 2.34 it isa polynomial map.

The affine coordinate ring of D(f) is

A(D(f)) = A(X)f = {f i}−1A(X).

This is because 1/f is a regular function on D(f).

Regular maps between quasi-projective varieties are continuous in the Zariski topology.

3.2 The Function Field of an Irreducible Quasi-Projective Variety

Let X ⊆ Pn be an irreducible quasi-projective variety. Recall that S = K[x0, . . . , xn] denotes the polynomialring in n + 1 variables, and Sd its homogeneous component of degree d. We denote homogeneous localizationat a prime ideal p by S(p). We write

OX :={FG

: F,G ∈ Sd for some d, and G /∈ I(X)}⊆ S(I(X))

This works, because X is irreducible (even if not necessarily closed), so the ideal I(X) is prime. In the localring OX we have the maximal ideal

MX ={FG∈ OX : F ∈ I(X)

}The function field of X is the quotient

K(X) := OX/MX .

As X is irreducible, every open subset U ⊆ X is dense, so I(X) = I(U), which implies K(X) = K(U). Inparticular, K(X) = K(X), where X ⊆ Pn denotes the projective closure of X. This means that when dealingwith function fields, we may reduce to the projective or affine case as we please.

Hartshorne gives another definition for the function field, which can be proven to be equivalent to the one above:

K(X) = {(U, f) : ∅ 6= U ⊆open X and f is a regular function on U}/ ∼,

where the equivalence relation is given by

(U, f) ∼ (V, g) ⇐⇒ f = g on U ∩ V.

Notice that U∩V 6= ∅ since X is irreducible. This also allows addition and multiplication to be properly defined:

(U, f) + (V, g) = (U ∩ V, f + g) and (U, f) · (V, g) = (U ∩ V, f · g).

Again because X is irreducible, we can see that K(X) is a field. Indeed, for (U, f) with f 6= 0 on X, then

(U, f)−1 = (U ∩D(f),1

f)

is well defined since U ∩D(f) 6= ∅.

16

3.3 Products of Quasi-Projective Varieties

3.3.1 The Segre Embedding.

Let m,n ∈ N. The product of two projective spaces is no longer a projective space. It is however still aprojective variety, and a product in the categorical sense. The structure of projective variety can be seen asfollows:

σ : Pn × Pm −→ P(n+1)(m+1)−1

Let N = (n+1)(m+1)−1. The coordinates on PN are xij with i = 0, . . . , n and j = 0, . . . ,m, and the equationsof σ(Pn×Pm are given by the 2−minors of the generic matrix. For any t = 1, . . . ,min{n,m} denote by It((xij))the ideal generated by the t−minors of the matrix (xij). To see that σ(Pn × Pm) = Z(I2((xij))), notice firstthat the inclusion from left to right is trivial. For the other inclusion, let P = (p00 : · · · : pnm) ∈ Z(I2((xij))).By the symmetry of the whole situation, one may assume without loss of generality that p00 6= 0, that is wemay assume p00 = 1. Then, using the equations of the 2-minors, we have:

1 p01 p02 . . . p0m

p10 p10p01 p10p02 . . . p10p0m

......

......

pn0 pn0p01 pn0p02 . . . pn0p0m

= σ((1 : p10 : · · · : pn0), (1 : p01 : · · · : p0m)).

Whenever we talk about Pn × Pm we do not mean is as a set, but we always think of it as its image under theSegre Embedding.

3.3.2 Products

If X ⊆ Pn and Y ⊆ Pm are quasi-projective varieties (i.e. locally closed), then X × Y ⊆ Pn × Pm is also locallyclosed in PN . This is the product X × Y as a quasi-projective variety. It is an easy check that this is actuallya categorical product as seen on Exercise Sheet 4. This means in particular that the maps πX : X × Y −→ Xand πY : X × Y −→ Y are regular.The Segre embedding is a natural choice, as it satisfies the natural properties one would want from a product.Check [Sha13, Section 5.1] for a discussion on this.While the definition above works just fine, it is sometimes practical to take a closer look at the type of equationsdefining closed sets in the product. We have see (Sheet 4, Exercise 5c), that the topology on Pn × Pm shouldnot be the product topology. Closed sets should be preimages of algebraic closed sets in P(n+1)(m+1)−1 =: PNunder the Segre map σ. Every closed subset of PN is the zero locus of finitely many homogeneous polynomialsin K[zij ]. For each such F (zij), homogeneous of degree d, the pullback is a polynomial

σ∗(F (zij)) = G(x0, . . . , xn; y0, . . . , ym) ∈ K[x0, . . . , xn; y0, . . . , ym],

which is homogeneous of degree d in both sets of variables; i.e. bihomogeneous of degree (d, d). Relaxing thisto bihomogeneous of degree (d, e) in K[x,y] gives the full description, as the next theorem shows.

Theorem 3.7. A subset X ⊆ Pn×Pm is a closed algebraic subset if and only if it is the zero locus of a systemof bihomogeneous equations

Gi(x0, . . . , xn; y0, . . . , ym) ∈ K[x,y]

Every closed algebraic subset of Pn × Am is the zero locus of a system of equations

gi(x0, . . . , xn;u1, . . . , um) ∈ K[x,u]

which are homogeneous in x.

17

But what is a bigrading exactly? In this specific case, the monomials form a K-vector space basis of S = K[x],and Mon(S) = {xa : a ∈ Nn+1} is a commutative unitary semigroup with respect to multiplication. Withthis in mind, the usual N-grading on S corresponds to the semigroup homomorphism deg : Mon(S) −→ N,with deg(xa00 . . . xann ) := a0 + · · ·+ an. Similarly, a bigrading on T := K[x,y] will correspond to the semigrouphomomorphism degbi : Mon(T ) −→ N2 given by

degbi(xayb) := (a0 + · · ·+ an, b0 + · · ·+ bm).

A polynomial G(x,y) ∈ T is then bihomogeneous of bidegree (d, e) ∈ N2 if all the monomials appearing in Gwith nonzero coefficient have bidegree (d, e). For instance x0x

21y

20 + x3

2y0y1 is bihomogeneous of degree (3, 2),but x3

0 + y30 is not bihomogeneous – it has two graded pieces of degrees (3, 0) and (0, 3) respectively.

Proof of Theorem 3.7. Every closed set of Pn × Pm is the preimage of a closed set in PN under the Segreembedding. So, by the discussion prior to the statement of the theorem, every closed set is given equationsG1 = · · · = Gr = 0, with each Gi bihomogeneous of degree (di, di) for some di ∈ N. To show that anybihomogeneous system of equations defines a closed set, it is enough to notice that if G is bihomogeneous ofdegree (d, e), with d < e, then

Z(G) = Z(xaG : deg(xa) = e− d).

The case e > d is completely analogous.To obtain the second part of the statement, it is enough to dehomogenize with respect to some variable yi.

3.3.3 Graphs

We have seen on Sheet 4, that t × P1 is a line, thus a subvariety in P1 × P1. This extends to the graphΓf := {(x, f(x)) : x ∈ X} of a regular map f : X −→ Y , which becomes also a subvariety of X × Y .

Lemma 3.8. Let f : X −→ Y be a regular map between quasi-projective varieties. The graph Γf is a closedsubset of X × Y .

Proof. This reduces easily to Y = Pm. Then let i : Pm −→ Pm be the identity. Consider then the regular map(f, i) : X×Pm −→ Pm×Pm with (x, y) 7−→ (f(x), y). Then we reduce to the diagonal being closed in the Segreembedding, which is something we saw on Sheet 4 in the special case of P1×P1. (see [Sha13, Lemma 1.4, p.57]for more details).

3.3.4 Fiber Products

See Harris p.30.

3.4 Linear Subspaces

An inclusion of K-vector spaces Kk+1 ' W ⊆ V ' Kn+1 induces a map P(W ) P(V ). The image L ofsuch a map is called linear subspace (or linear subvariety) of dimension k of P(V ) = Pn. When k = n − 1,L is called a hyperplane, and when k = 1, a line. Just as a W is defined by n − k linear equations, a linearsubvariety of dimension k is the zero locus of n−k linearly independent homogeneous polynomials of degree one(i.e. of linear forms). Conversely, any projective algebraic set which can be defined as the zero locus of linearforms is a linear subvariety in the sense of the above definition.The intersection of two linear subspaces is again a linear subspace (corresponding to the intersection of thedefining K-vector subspaces). The span of two linear subspaces L1 = P(W1) and L2 = P(W2) is the linear

18

subspace SpanP(L1, L2):= P(W1 + W2). It is the smallest linear subspace containing both L1 and L2. Usingthis point of view, the definition can be extended to any subset A of Pn:

SpanP(A) := the smallest linear subspace of Pn containing A.

So SpanP(L1, L2) = SpanP(L1 ∪L2). Using the dimension formula for the sum of K-vector spaces, one recoversthe same relation for the linear subspaces of Pn:

dim SpanP(L1, L2) = dimL1 + dimL2 − dim(L1 ∩ L2).

This implies that in the projective space Pn, if dimL1 + dimL2 ≥ n, then dim(L1 ∩L2) ≥ dimL1 + dimL2−n,in particular L1 ∩ L2 6= ∅. This is the generalization of any two lines intersect in P2 to higher dimensionalprojective spaces.

3.5 Finite sets of Points

One point is a particular(ly boring) case of linear space. However, more points at once, but still finitely many,can be interesting. There are a lot of things to say about finite sets of points in projective space. First of all,such sets are always closed. Here is a teaser for more.

Points pi ∈ Pn are equivalence classes of vectors vi ∈ V \{0}. We say that p1, . . . , pr ∈ Pn are independent if thecorresponding vectors v1, . . . , vr are linearly independent in V . This is clearly independent of the representatives.Equivalently, the points are independent if dim SpanP(p1, . . . , pr) = r − 1. Independence makes sense only forup to n+ 1 points, because any n+ 2 points in Pn are dependent. However, for more than n+ 1 points we havethe following concept. The points p1, . . . , pr ∈ Pn are said to be in general position if no n + 1 of them aredependent. If r ≥ n+ 1, this is equivalent to no n+ 1 of them lie in a hyperplane. A finite collection of pointsis a closed set. So one can look at the degrees of the equations defining this algebraic set. This says somethingabout the degrees of the curves, surfaces, hypersurfaces containing the given set of points. Here is one result inthis direction:

Theorem 3.9. If Γ ⊆ Pn is a collection of d ≤ 2n points in general position, then Γ is the zero locus of a finiteset of quadratic polynomials.

Proof. Check [Har92, Theorem 1.4]

Let us restrict now to points in the projective plane P2. We are interested in the smallest degree of a curvecontaining a given set of points. Of course there are particular configurations: for instance the points may becollinear, so the degree would be one in this case. We want to consider this question in a generic setting. For themoment think of “generic points” as “random points”. A curve C in P2 is a hypersurface, that is it is defined asthe zero locus of a single homogeneous polynomial F . So for C = Z(F ) to contain a finite collection Γ of pointsof P2 it means that F ∈ I(Γ). The simplest case is that of two points: through any two generic points therepasses one line. The next case is that of three points. Three points they are generic it they are not collinear.In this case, the smallest degree of a curve passing through them is two. One can show that the same holds forfour and five points, but that six generic points do not lie on a conic3. The answer to our question is not hardto find:

Theorem 3.10. Let Γ ⊆ P2 be a collection of m generic points. Let d ∈ N be the unique natural number with(d+ 1

2

)≤ m <

(d+ 2

2

).

Then the smallest degree of a curve containing Γ is d.3i.e. curve of degree two.

19

Proof. A plane projective curve of degree d is defined by a homogeneous polynomial of degree d in three variables.Such a polynomial F is a sum of

(3+d−1d

)=(d+2

2

)monomials. Let Γ = {p1, . . . , pm}. The curve C = Z(F )

contains Γ if and only ifF (p1) = · · · = F (pm) = 0.

This is a linear system of m equations with(d+2

2

)unknowns – the coefficients of the polynomial. What we want

is for the system to have a nontrivial solution. Now we can also make “generic” precise: the points are genericif the matrix of this homogeneous linear system is of maximal rank. The conclusion then follows.

While this proof is elementary, a natural generalization of this theorem is still a widely open conjecture, knownas Nagata’s Conjecture on Curves4. We will not make a precise statement of it here, but the idea is the following:If we ask for the curve to pass through each point pi with a given multiplicity mi, then there is a natural guessfor this minimal degree as well. The conjecture says that this guess is right.

3.6 The Grassmannian

3.6.1 The exterior product of two vectors

Let us recall the exterior product of a finite dimensional K-vector space V . An alternating bilinear form is amap B : V × V −→ K satisfying the axioms:

(Bil 1) B(v, w) = −B(w, v),

(Bil 2) B(λ1v1 + λ2v2, w) = λ1B(v1, w) + λ2B(v2, w),

for all λ• ∈ K and v•, v, w ∈ V . We can add alternating bilinear forms and multiply them with scalars andget a thus the K-vector space of bilinear forms on V . This space is isomorphic to the space of skew-symmetricmatrices of the appropriate size. The second exterior power

∧2V of V is the dual to the space of alternating

bilinear forms on V . The exterior product of v and w is (v ∧ w) ∈∧2

V given by

(v ∧ w)(B) := B(v, w).

We have the obvious properties:

(i) v ∧ w = −w ∧ v,

(ii) (λ1v1 + λ2v2) ∧ w = λ1v1 ∧ w + λ2v2 ∧ w,

(iii) If {v1, . . . , vn} is a basis of V , then {vi ∧ vj : 1 ≤ i < j ≤ n} is a basis of∧2

V .

Furthermore: v ∧ w = 0 if and only if v = λw for some λ ∈ K.

3.6.2 The relation with two-dimensional subspaces

We are interested in the exterior product because every two-dimensional subspace of V corresponds to a one-dimensional subspace of

∧2V . Take U = SpanK{v, w} ⊂ V with dimK U = 2. Then every other basis {u1, u2}

of U is obtained from {v, w} via an invertible matrix A =

(a bc d

)∈ GL2(K) as u1 = av+bw and u2 = cv+dw.

u1 ∧ u2 = (av + bw) ∧ (cv + dw)

= ad(v ∧ w) + bc(w ∧ v)

4You may check Wikipedia on this, or the paper of Ciliberto, Harbourne, Miranda, and Roé :https://arxiv.org/abs/1202.0475

20

= det(A)(v ∧ w).

So, SpanK{v ∧ w} ⊆∧2

V is independent of the choice of basis of U . Thus, to every 2-dimensional subspaceof V corresponds a point in P(

∧2V ). That is why it is good to think about projective spaces as associated to

vector spaces, and not simply as Pn. In this case, just writing Pn(n−1)

2 would have been less illuminating.The first question that comes up is: do all 1-dimensional subspaces of Λ2V correspond to a 2-dimensionalsubspace of V ? The answer is: No, not every element of

∧2V is of the form v∧w. For instance, v1∧v2 +v3∧v4,

where {v1, . . . , v4} is a basis of V . (This is an easy linear algebra exercise). So, the association U 7−→ v ∧ wgives a subset of the projective space P(

∧2V ). Is this algebraic? We will see that it is, and that this holds

more generally.

3.6.3 Exterior powers

To parametrize the collection of subspaces of any dimension with a projective algebraic set, we have too take alook at alternating multilinear maps M : V k −→ W . A multilinear map 5 is alternating, if M(v1, . . . , vk) = 0whenever there exists i 6= j with vi = vj . The determinant, viewed as a map det : (Kn)n −→ K is an alternatingmultilinear form. Also the cross product ϕ : (K3)2 −→ K3, given by

ϕ((a, b, c), (x, y, z)) := (bz − cy, cx− az, ay − bx),

is an alternating bilinear map.

The k-th exterior power of V , also known as the alternating k-fold tensor product, is given by the uniquepair (

∧kV, τk), where

∧kV is a k-vector space and τk : V k −→

∧kV is an alternating k-linear map, satisfying

the following universal property: For every k-fold alternating linear map f : V k −→ W , there exists a uniquelinear map g :

∧k −→W such that f = g ◦ τk. That is such that the following diagram commutes:

V k W

∧kV

f

τkg

One can easily see that∧k

V = V ⊗k/L, where L = SpanK{v1 ⊗ · · · ⊗ vk : vi = vj for some i 6= j}.

Remark 3.11. Assume that {e1, . . . , en} is a basis of V . A basis of∧k

V is given by the alternating tensors:

{ei1 ∧ . . . ∧ eik : 1 ≤ i1 < · · · < ik ≤ n}.

So dimK∧k

V =(nk

). In particular,

∧0V = K,

∧1V = V and

∧nV = SpanK(e1 ∧ · · · ∧ en). All higher exterior

powers vanish.

The exterior algebra of the n-dimensional K-vector spaces V is∧V :=

⊕ni=0

∧iV , with product operation

induced by(v1 ∧ . . . ∧ vi) ∧ (w1 ∧ . . . wj) := v1 ∧ . . . ∧ vi ∧ w1 ∧ . . . ∧ wj ∈

∧i+jV,

∀ v1 ∧ . . . ∧ vi ∈∧i

V and w1 ∧ . . . ∧ wj ∈∧j

V.

Remark 3.12. Let V = Kn and vi = (ai,1, . . . , ai,n) for i = 1, . . . , k. Then, we can express the coordinates ofv1 ∧ · · · ∧ vk with respect to the canonical basis in terms of the k-minors of (aij):

v1 ∧ · · · ∧ vl =∑

j1,...,jk

a1,j1 . . . ak,jk · ej1 ∧ · · · ∧ ejk

5that is linear in each argument.

21

=∑

1≤j1<···<jk≤n

(∑σ∈Sk

sign(σ)a1,σ(j1) . . . ak,σ(jk)

)· ej1 ∧ · · · ∧ ejk

=∑

1≤j1<···<jk≤n

det(Aj1,...,jk) · ej1 ∧ · · · ∧ ejk ,

where Aj1,...,jk is the k × k submatrix of (aij) with the columns j1, . . . , jk. For example, if k = 2, n = 3, then

(aij) =

(a11 a12 a13

a21 a22 a23

)and

v1 ∧ v2 = (a11a22 − a12a21)e1 ∧ e2 + (a11a23 − a13a21)e1 ∧ e3 + (a12a23 − a13a22)e2 ∧ e3.

So the exterior product encodes all the maximal minors of a matrix in one object. As a consequence we have:

v1 ∧ · · · ∧ vk = 0⇔ v1, . . . , vk are linearly dependent.

Furthermore, if the two sets {v1, . . . , vk} ⊆ V and {w1, . . . , wk} ⊆ V are both linearly independent, then

SpanK(v1, . . . , vk) = SpanK(w1, . . . , wk) ⇐⇒ v1 ∧ · · · ∧ vk = λ · w1 ∧ · · · ∧ wk for some λ ∈ K. (3.1)

This last equivalence needs some proof, but it’s all basic linear algebra.

3.6.4 The Plücker Embedding

Definition 3.13. Let n ∈ N>0 and k ∈ N with 0 ≤ k ≤ n. The Grassmannian of k-planes in the n-dimensional K-vector space V is the set of all k-dimensional linear subspaces of V . We denote this by Gr(k, V ).When V = Kn, we simply write

Gr(k, n) = {W ⊆ Kn : dimKW = k}.

For k = 1, we obtain Pn−1 (as a set for the moment). For any k ≥ 1, by the correspondence between k-dimensional subspaces of V and k − 1 linear subspaces of P(V ), we have a bijection between the collection ofk-vector subspaces of Kn and k − 1-dimensional linear subspaces of Pn−1. For this reason, one may find thenotation Gr(k − 1, n− 1) for the same object. In order to avoid confusion, we will denote by

G(k − 1, n− 1) := {L ⊆ Pn−1 : L is a linear subspace of dimension k − 1}.

The Plücker Embedding is the map P` : Gr(k, V ) P(∧k

(V )) given by

P`(SpanK(v1, . . . , vk)) := [v1 ∧ · · · ∧ vk] ∈ P(

k∧(V )).

By (3.1) this is well defined and injective. For a k-dimensional subspace U ⊆ V , the homogeneous coordinates ofP`(U) in P(

∧kV ) are called the Plücker coordinates of U . Once a basis of V is chosen (thus an isomorphism

with Kn and a corresponding basis of∧k

V as well), the Plücker coordinates are just the maximal (i.e. k−)minors of the matrix with the coordinates of v1, . . . , vk as rows.

Example 3.14. (a) For k = 1, and U = SpanK(a1e1 + · · ·+ anen) we get P`(U) = (a1 : . . . : an) ∈ Pn−1.

(b) Now take U = SpanK(e1 + e2, e2 + e3) ∈ Gr(2, 3). We have

(e1 + e2) ∧ (e1 + e3) = −e1 ∧ e2 + e1 ∧ e3 + e2 ∧ e3,

so P`(U) = (−1 : 1 : 1).

22

To show that the image of the Plücker embedding is a projective algebraic set we need to express being a puretensor, that is being of the form v1∧· · ·∧vk ∈

∧kV , as a polynomial condition. For this we need the following

lemma.

Lemma 3.15. Let 0 6= w ∈∧k

V , with k < n, and define fw : V −→∧k+1

V by

fw(v) := v ∧ w.

Then rank fw ≥ n− k, and, most importantly,

rank fw = n− k ⇔ ∃ v1, . . . , vk ∈ V with w = v1 ∧ · · · ∧ vk.

Proof. Let r = dim Ker fw = n− rank fw. Choose a basis {v1, . . . , vr} of fw, extend it to a basis {v1, . . . , vn} ofV , and express w =

∑i1<···<ik pi1...ikvi1 ∧ · · · ∧ vik . Then use the fact that for i = 1, . . . , r we have vi ∧ w = 0.

This implies that only the pi1...ik with {1, . . . , r} ⊆ {i1, . . . , ik} may be nonzero. This gives the inequality r ≤ k,which is equivalent to rank fw ≥ n− k.For the second part, clearly if r = k, then w = λ · v1 ∧ · · · ∧ vr. In the other direction, if w = w1 ∧ . . . ∧ wk, asw 6= 0 the wi are linearly independent, and they belong to Ker fw. By r ≤ k, we are forced to have the requiredequality.

Proposition 3.16. The image of the Plücker embedding is an algebraic subset of P(∧k

V ).

Proof. For k = n we just have one point. If k < n, then [w] ∈ P(∧k

V ) is in P`(Gr(k, V )) if and only if w is apure tensor in

∧kV . By Lemma 3.15 this is equivalent to rank fw : V −→

∧k+1V having rank n−k. Choosing

a basis, the matrix of this linear map will be a(nk+1

)× n matrix, with entries the homogeneous coordinates of

[w] (with some repetitions and permutations). As the rank is in general at least n − k, the rank condition isgiven by the vanishing of the (n− k + 1)−minors of the corresponding matrix. These minors are homogeneouspolynomials in the entries of the matrix, and thus in the homogeneous coordinates of w.

Example 3.17. Let’s take a look at the specific case Gr(2, 4) ⊆ P5. Denote the coordinates on P5 by

x12, x13, x14, x23, x24, x34.

Let [w] = (p12 : p13 : p14 : p23 : p24 : p34) ∈ P5. The corresponding element of∧2

V is

w = p12 · e1 ∧ e2 + p13 · e1 ∧ e3 + p14 · e1 ∧ e4

p23 · e2 ∧ e3 + p24 · e2 ∧ e4 + p34 · e3 ∧ e4.

This means that fw maps the basis elements e1, e2, e3, e4 as follows:

e1 7−→ p23 · e1 ∧ e2 ∧ e3 + p24 · e1 ∧ e2 ∧ e4 + p34 · e1 ∧ e3 ∧ e4

e2 7−→ −p13 · e1 ∧ e2 ∧ e3 − p14 · e1 ∧ e2 ∧ e4 + p34 · e2 ∧ e3 ∧ e4

e3 7−→ p12 · e1 ∧ e2 ∧ e3 − p14 · e1 ∧ e3 ∧ e4 − p24 · e2 ∧ e3 ∧ e4

e4 7−→ p12 · e1 ∧ e2 ∧ e4 + p13 · e1 ∧ e3 ∧ e4 + p23 · e2 ∧ e3 ∧ e4

So, the matrix which is supposed to have rank 2 isp23 −p13 p12 0p24 −p14 0 p12

p34 0 −p14 p13

0 p34 −p24 p23

23

Computing the sixteen 3−minors of the above matrix, (with the pij replaced by variables), we obtain thegenerators:

g1 = x214x23 − x13x14x24 + x12x14x34, g2 = x14x23x24 − x13x

224 + x12x24x34,

g3 = x14x23x34 − x13x24x34 + x12x234, g4 = 0

g5 = −x13x14x23 + x213x24 − x12x13x34, g6 = −x14x

223 + x13x23x24 − x12x23x34,

g7 = 0 g8 = x14x23x34 − x13x24x34 + x12x234,

g9 = x12x14x23 − x12x13x24 + x212x34, g10 = 0

g11 = −x14x223 + x13x23x24 − x12x23x34, g12 = −x14x23x24 + x13x

224 − x12x24x34,

g13 = 0 g14 = x12x14x23 − x12x13x24 + x212x34,

g15 = x13x14x23 − x213x24 + x12x13x34, g16 = x2

14x23 − x13x14x24 + x12x14x34.

Once the matrix M is defined, the Macaulay2 command to obtain this list of minors is exteriorPower(3,M).The command minor(3,M) returns the ideal generated by these minors. There seems to be some pattern tothis, namely if f = x14x23 − x13x24 + x12x34, then

g1 = x14 · f g2 = x24 · f g3 = x34 · fg5 = −x13 · f g6 = −x23 · f g8 = x34 · fg9 = x12 · f g11 = −x23 · f g12 = −x24 · fg14 = x12 · f g15 = x13 · f g16 = x14 · f

This means, that the ideal generated by the 3−minors is

I = 〈x12, x13, x14, x23, x24, x34〉 ∩ 〈x14x23 − x13x24 + x12x34〉 .

Which means IP(P`(Gr(2, 4))) = 〈x14x23 − x13x24 + x12x34〉. We will talk about this trick later. So theGrassmannian of planes in four-space is a quadrics in P5.

Let us now have a look at the affine patches of the Grassmannian. These will shed some light on the structureof this variety. In particular, they provide an easy way to compute the dimension.

3.6.5 Affine Cover of the Grassmannian

We will from now on identify Gr(k, n) with its image under the Plücker embedding, and write Gr(k, n) ⊆P(n

k)−1. We view a point of P ∈ Gr(k, n) both as a point in projective space, with projective coordinates(pi1...ik)1≤i1<···<ik≤n, and as a vector subspace of Kn. Each coordinate corresponds to the coefficient of ei1 ∧· · · ∧ eik ∈

∧k Kn. Let U0 := {P ∈ Gr(k, n) : p1...k 6= 0}. By Remark 3.12 this means that the vector space Pis generated by the rows of a k× n matrix, with the first minor (the one with column indices 1, . . . , k) nonzero.Let the matrix be (

A | B)

with A ∈ GLk(K) and B ∈ Matk,n−k(K). For each such vector space, multiplying the matrix with A−1 (to theleft), does not change the row space, that is our point P , but brings the matrix to the form 1 0

. . . A−1B0 1

(3.2)

Viewing Matk,n−k(K) as a(k · (n− k)

)-dimensional affine space, we have a map:

ϕ0 : Ak(n−k) U0

M row span of(Ik | C

)

24

The correspondence between the matrix presentation of a subspace and the Plücker coordinates is obtainedby taking minors, which are polynomials in the entries of C. So the map ϕ0 above gives us a morphism.Furthermore, two difference matrices C and C ′ define different subspaces, so ϕ0 is also bijective. The inverse ofϕ0 is defined by

U0 3 (pi1,...,ik)1≤i1<···<ik≤n 7−→ ((−1)i+j+k−1p1,...,i,...,k,j)i,j ∈ Matk,n−k(K) = Ak(n−k).

So ϕ−1 is also a morphism, and we conclude that U0 is isomorphic to Ak(n−k). We have just proved the following.

Proposition 3.18. The Grassmannian Gr(k, n) can be covered with finitely many affine patches isomorphic tothe affine space Ak(n−k). In particular, the dimension of Gr(k, n) is k(n− k).

Every vector subspace of dimension k of An is given by a k×n matrix of maximal rank. Two matrices give thesame subspace if they are equivalent modulo row transformations. So a canonical representative can be chosenas a matrix in reduced row echelon form. This gives a presentation of Gr(k, n) as a disjoint union of affinespaces. For instance, for k = 2, n = 4 we have the following possible reduced row echelon shapes:(

1 0 ∗ ∗0 1 ∗ ∗

) (1 ∗ 0 ∗0 0 1 ∗

) (1 ∗ ∗ 00 0 0 1

)(

0 1 0 ∗0 0 1 ∗

) (0 1 ∗ 00 0 0 1

) (0 0 1 00 0 0 1

).

That is, Gr(2, 4) is the disjoint union A4 t A3 t A2 t A2 t A1 t A0.

Remark 3.19. Assuming that the Plücker coordinate p1,...,k = 1, we call the others the affine (or local)coordinates in U0. These correspond now to all the minors of all sizes the right-most (k × (n− k))− block (i.e.to A−1B in (3.2)). By Laplace expansion of each determinant we thus get only quadratic relations among thePlücker coordinates, which are called the Plücker relations. For instance, in Gr(2, 4) ∩ U0 we get(

1 0 x1,3 x1,4

0 1 x2,3 x2,4

)And the right-most 2−minor is the coordinate x3,4. Expanding this we get

x3,4 = x1,3x2,4 − x1,4x2,3.

To recover the homogeneous Plücker relations we have to homogenize with respect to x1,2, and we obtain theequation f from Example 3.17.

A final result, without proof: Gr(k, n) ' Gr(n− k, n). The bijection is clear: take the orthogonal complementof the subspace. One just needs to check that this gives a morphism of quasi-projective varieties.

Consider watching the video (35’) now.

3.7 Regular Maps

Let X be a quasi-projective variety. We will present two ways to define regular maps, the first is better suited forgeneralizations, the second more practical. The first definition is the following. A map ϕ : X −→ Pn is a regularmap, if each affine piece is regular. That is if it is continuous, and for each affine piece An ' Ui = D(xi) ⊆ Pnthe restriction ϕ : ϕ−1(Ui) −→ Ui is an affine regular map, and thus by Theorem 2.34, a polynomial map.

25

It would be better to define regular maps as (n+ 1)-tuples of regular functions on X, but this is too restrictiveif X is projective. However, we may specify a tuple of homogeneous polynomials of the same degree, with thecondition that they do not vanish simultaneously at any point of X. But this is still too restrictive: [Harris p21].

In contrast to the affine case, two projective varieties X and X ′ being isomorphic is not equivalent to isomorph-isms of the corresponding projective coordinate rings S(X) and S(X ′). The latter is stronger, and is calledprojectively equivalent, that is there is a projective transformation mapping X to X ′. An example where thetwo differ is the Veronese embedding.

Recall PGLn+1(K) = GLn+1(K)/K∗, whose elements are called projective (linear) transformations. It acts onPnK by sending (a0 : · · · : an)T 7−→ M · (a0 : · · · : an)T , which preserves collinearity. We will see later (we needBezout) that Aut(Pn) = PGLn+1(K). For the moment, recall that: For any two sets of n+ 2 points in generalposition in Pn: {p1, . . . , pn+2} and {q1, . . . , qn+2} there exists exactly one projective (linear) transformationmapping pi to qi for all i.

The image of an affine closed set under a regular map may be neither open or closed. We have seen this inExercise 3 on Sheet 2. Most importantly, closed sets are not always mapped to closed sets. An example of thisis the inclusion of An in Pn. Projective varieties behave much better in this sense:

Theorem 3.20. The image of a projective variety under a regular map is closed.

Proof. It follows from the next theorem and Lemma 3.8.

Theorem 3.21. If X is a projective variety and Y is a quasi-projective variety, then the projection on thesecond factor πY : X × Y −→ Y is a closed map.

Proof. So our goal is to show that, if Z ⊆closed X × Y , then πY (Z) ⊆closed Y . In the first part of this proof wewill show that it is enough to prove the theorem for πAm : Pn × Am −→ Am.

• X is a projective variety, thus X ⊆closed Pn. This means that X × Y ⊆closed Pn × Y . So, if Z ⊆closedX × Y ⇒ Z ⊆closed Pn × Y , and we may thus replace X with Pn for our proof.

• By Lemma 3.3 being closed is a local property. So we may cover Y with affine open sets Ui, which wecan choose by Lemma 3.4 to be affine varieties. So we can reduce to Pn×Ui, with Ui an affine variety (inShafarevich’s sense).

• Using the isomorphism Ui ' Yi ⊆closed Am we may replace Pn ×Ui with Pn × Yi, with Yi an affine closedset.

• Finally, similarly to the first step, as Pn × Yi ⊆closed Pn × Am we are allowed to restrict to Pn × Am.

By the second part of Theorem 3.7, closed sets Z ⊆ Pn × Am are of the form

Z = Z(g1(x0, . . . , xn;u1, . . . , um), . . . , gt(x0, . . . , xn;u1, . . . , um))

where each gi is homogeneous in x. For a point a = (a1, . . . , am) ∈ Am we have π−1Am(a) = {(p, a) ∈ Pn × Am :

gi(p, a) = 0}. In particular,p ∈ πAm(Z) ⇐⇒ ZPm(gi(x, a)) 6= ∅

This means that the statement we want to prove is

T := {a ∈ Am : g1(x, a) = · · · = gt(x, a) = 0 has a nontrivial solution in x} is a closed subset of Am.

By the projective Nullstellensatz, we have

T = {a ∈ Am : ms 6⊆ (g1(x, a), . . . , gt(x, a)), ∀s ∈ N},

26

where m ⊆ K[x] is the irrelevant maximal ideal. For every s ∈ N we denote by Ts = {a ∈ Am : ms 6⊆(g1(x, a), . . . , gt(x, a))}. As T =

⋂s∈N Ts, it will be enough to prove that each Ts is closed.

Let Mon(K[x])s = {Mα}α be the set of monomials in x of degree s. We have that ms = 〈K[x]s〉. So ms ⊆〈g1(x, a), . . . , gt(x, a)〉 is equivalent to

〈g1(x, a), . . . , gt(x, a)〉s := 〈g1(x, a), . . . , gt(x, a)〉 ∩K[x]s = K[x]s.

Now, let di := degx(gi) for every i = 1, . . . , t. As a K-vector space, 〈g1(x, a), . . . , gt(x, a)〉s is generated by

{gi(x, a) ·Mβi : β ∈ Nn+1 with degMβ

i = s− di}i=1,...,t.

Now write down each of the above generators as a column vectors with respect to the monomials of degree s inx0, . . . , xn, and collect them in a matrix (see Example 3.22 after the proof). The condition for these generatorsnot to span the whole K[x]s is equivalent to said matrix not having maximal rank, which is equivalent to thevanishing of all the maximal minors. This is obviously an algebraic condition, and we conclude.

Example 3.22. Let us take n = 1,m = 2, that is look at closed sets in P1 × A2. Let us take the specificexample of Z defined by the bihomogeneous equations

g1(x,u) = x20u1 + x2

1u2

g2(x,u) = x0u31 + x0u1u

22 + x1u

31.

For s = 1, there is nothing to check because there is just one 1-form. For s = 2 we have

T2 = {(a1, a2) ∈ A2 : m2 6⊆ 〈g1(x; a1, a2), g2(x; a1, a2)〉}

and for this we need that the matrix below has to have rank less than three: u1 0 u2

u31 + u1u

22 u3

1 00 u3

1 + u1u22 u3

1

This gives us the equation:

f := u71 + u6

1u2 + 2u41u

32 + u2

1u52 = 0

When s = 3, we look at

T3 = {(a1, a2) ∈ A2 : m3 6⊆ (g1(x; a1, a2), g2(x; a1, a2)}= {a ∈ A2 : dimK 〈x0g1(x;a), x1g1(x;a), x2

0g2(x;a), x0x1g2(x;a), x21g2(x;a)〉 < dimK K[x]3 = 4.

We write down the five generators with respect to the basis {x30, x

20x1, x0x

21, x

31} of K[x]3:

x0g1(x;u) = u1 · x30+ 0 · x2

0x1+ u2 · x0x21+ 0 · x3

1

x1g1(x;u) = 0 · x30+ u1 · x2

0x1+ 0 · x0x21+ u2 · x3

1

x20g2(x;u) = (u3

1 + u1u22) · x3

0+ u31 · x2

0x1+ 0 · x0x21+ 0 · x3

1

x0x1g2(x;u) = 0 · x30+ (u3

1 + u1u22) · x2

0x1+ u31 · x0x

21+ 0 · x3

1

x21g2(x;u) = 0 · x3

0+ 0 · x20x1+ (u3

1 + u1u22) · x0x

21+ u3

1 · x31

This means that

T3 ={

(a1, a2) ∈ A2 : rank

u1 0 u2 00 u1 0 u2

u31 + u1u

22 u3

1 0 00 u3

1 + u1u22 u3

1 00 0 u3

1 + u1u22 u3

1

≤ 3}

27

So the equations of T3 are

u71u2 + u6

1u22 + 2u4

1u42 + u2

1u62 = 00 = 0

u81 + u7

1u2 + 2u51u

32 + u3

1u52 = 0

u101 + u9

1u2 + 2u71u

32 + u5

1u52 = 0

−u101 − u9

1u2 − u81u

22 − 3u7

1u32 − 3u5

1u52 − u3

1u72 = 0

An easy check shows that they are all multiples of the equation f = 0 defining T2. So T2 ⊆ T3. One can checkthat this goes on for all Ti, and in the end we find T = T2.

Theorem 3.20 also holds for another important class of maps: proper maps. These are the algebraic versionof the topological notion: proper maps are those morphisms for which the inverse image of a compact set iscompact. We will say that a map between quasi-projective varieties f : X −→ Y is a proper morphism ifit factors as the composition of a closed embedding i : X ↪→ Pn × Y and a projection p : Pn × Y −→ Y . Iff : X −→ Y is a proper map, then f−1(y) is a projective variety for every y ∈ Y . If f : X −→ Y is a regularmap of projective varieties, then the restriction f : f−1(U) −→ U to an open set of Y is proper.

3.7.1 Projections

Let P = Z(L1, . . . , Lr) be a linear subspace of Pn of dimension n− r. The stereo graphic projection with centerP is the rational map πP : Pn −→ Pr−1 is given by

πP (x0 : · · · : xn) := (L1(x0, . . . , xn) : · · · : Lr(x0, . . . , xn)).

This map is regular everywhere on Pn \ P . If X ⊆ Pn \ P , then one has a regular map πP : X −→ Pr−1.

3.7.2 Consequences of being closed

We are now getting at an important difference between affine and projective varieties: functions that are regulareverywhere on an irreducible projective variety must be constant.

Theorem 3.23. If ϕ is a regular function on an irreducible closed projective set, then ϕ is constant.

Proof. Let X be the closed irreducible projective set. The regular function ϕ : X −→ K can be seen as a mapϕ : X −→ A1, which we can further see as a regular map ϕ : X −→ P1. By Theorem 3.20 the image ϕ(X) ⊆ P1

is closed in P1 and also a subset of A1. So it has to be a finite set: ϕ(X) = {p1, . . . , pr}. But if r ≥ 1, thenX = ϕ−1(p1) ∪ · · · ∪ ϕ−1(pr) would contradict the irreducibility of X. So ϕ(X) = p, i.e. ϕ is a constant.

We get as an immediate consequence the following.

Corollary 3.24. A regular map f : X −→ Y from an irreducible projective variety X to an affine variety Ymaps X to a point.

A more interesting consequence of Theorem 3.20 is a sort of estimate of “how many” polynomials are irreducible.Namely, we are going to see that “random” forms over algebraically closed fields are irreducible. Let’s be moreprecise: A form of degree d in n + 1 variables is a nonzero vector in K[x0, . . . , xn]d, which is a K-vector spaceof dimension N + 1 =

(n+dd

). As we are interested in the zero locus of forms and multiplying with a nonzero

scalar does not change that, we regard forms as points in PN , and every point in PN corresponds to a form. =

Proposition 3.25. The set X = {F ∈ Pn : F is reducible } is a closed subset.

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Proof. Let us split X as a finite union. For every k = 1, . . . , bd2c let

Xk = {F ∈ Pn : ∃ G1, G2 forms with degG1 = k and F = G1 ·G2}.

Clearly we have X =⋃d/2k=1Xk, where the union is not disjoint. It is thus enough to prove that each Xk is

closed. Consider now Mk + 1 =(n+kk

)and the map

fk : PMk × PMd−k −→ PN

which sends two forms to their product. This is clearly6 regular. So, as in general Pm × Ps is a projectivevariety, and thus, Xk = im fk is closed by Theorem 3.20.

Remark 3.26. For quadrics you have seen this in linear algebra. Over fields of characteristic different fromtwo, forms of degree two are given by symmetric matrices, with their coefficients appearing as entries in thematrix, and a quadrics is degenerate when the determinant is zero.

Proposition 3.25 gives a similar statement for non necessarily homogeneous polynomials of degree ≤ d: the setof reducible polynomials of degree d together with the set of polynomials of degree < d form a closed set.

3.7.3 Finite Maps

Let f : X −→ Y be a regular map between affine varieties. If f(X) is dense in Y , then we have an inclusion ofK-algebras f∗ : A(Y ) ↪→ A(X), meaning that we can view A(X) as an A(Y )-algebra. We say that f is a finitemap if A(X) is integral over A(Y ).

Remark 3.27. 1. The composition of finite maps is a finite map.

2. The projection f : A2 −→ A1, restricted to Z(xy − 1) is not finite!? Why?

3. Projections X −→ X/G to quotient spaces by actions of finite groups are finite (leave this as homework?)

4. The inverse image of a point under a finite map is at most finite. To see this, it is enough to show thatthe coordinate functions take only finitely many values. But the coordinate functions are integral overA(Y ). [Expand]

Theorem 3.28. Every finite map is surjective.

Proof. Let f : X −→ Y be the finite map, and q = (q1, . . . , qn) ∈ Y ⊆ An. Let mq be the maximal ideal of qin A(Y ) = K[y1, . . . , yn]/IY , that is, it is the ideal mq = (y1 − q1, . . . , yn − qn)/IY . So we have

f−1(q) = Z(f∗(y1)− q1, . . . , f∗(yn)− qn).

This means that f−1(q) = ∅ if〈f∗(y1)− q1, . . . , f

∗(yn)− qn〉 = A(X)

Viewing A(X) as an A(Y )−module via f∗, and identifying mq with its image under f∗, the above conditionreduces to mq ·A(X) = A(X). As f is finite, the A(Y )−module A(X) is finitely generated, and we conclude by(a corollary of) Nakayama’s Lemma (see [Sha13, Lemma on p.61, Corollary A.1]).

Corollary 3.29. Finite maps are closed.

Proof. Let f : X −→ Y be a finite map, and Z ⊆ X a closed set. It is enough to prove that f(Z) is closed whenZ is also irreducible. Regard the restriction f : Z −→ f(Z), which is also a finite morphism of affine varieties.Thus, by Theorem 3.28, f(Z) = f(Z).

6If it is not clear, try writing an example down for small values of n and d.

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Finiteness is a local property. More precisely:

Theorem 3.30. If f : X −→ Y is a regular map of affine varieties, and for every point q ∈ Y , there exists anopen neighborhood U with q ∈ U ⊆ Y , such that the restriction f : f−1(U) −→ U is finite, then f is also finite.

Proof. See [Sha13, p62, Theorem 1.13].

Definition 3.31. A regular map of quasi-projective varieties f : X −→ Y is finite if any point y ∈ Y has anaffine neighborhood U such that f−1(U) is affine and f : f−1(U) −→ U is a finite map between affine varieties.

Theorem 3.32. If f : X −→ Y is a regular map with f(X) dense in Y , then f(X) contains an open subset ofY .

The previous theorem shows that regular maps of algebraic varieties are more restrictive than continuous ordifferentiable maps. For the latter, you may have a line that is everywhere dense on a torus. But somethinglike this may not happen for regular maps.

Theorem 3.33. Stereographic projections πP : Pn \ P −→ Pr−1, with P a linear subspace of dimension nrestricted to a closed subset X ⊆ Pn with X ∩ P = ∅ give a finite map X −→ πP (X).

Theorem 3.34. Let X ⊆ Pn be a closed subset. If F0, . . . , Fs ∈ K[x0, . . . , xn]d are forms of degree d which donot vanish simultaneously on X, then the map ϕ : X −→ ϕ(X) ⊆ Ps defined by

ϕ(p) = (F0(p) : · · · : Fs(p))

is a finite map.

Proof. The theorem follows from Theorem 3.33 after applying a Veronese embedding of degree d of X. Theforms are mapped to linear forms L0, . . . , Ls in

(n+dd

)variables, and our map factors as ϕ = π ◦ vd, where π is

the stereographic projection with center Z(L0, . . . , Ls).

3.7.4 Noether Normalization

Recall the theorem from commutative algebra.

Theorem 3.35 (Noether normalization). Let K be an infinite field and A = K[a1, . . . , an] be a finitely generatedK-algebra. Then there exist y1, . . . , ym ∈ A, with m ≤ n such that

(a) y1, . . . , ym are algebraically independent over K, and

(b) A is a finite K[y1, . . . , ym]-algebra.

The algebraic theorem has a nice geometric consequence.

Theorem 3.36. For any irreducible projective variety X, there exists a finite map to a projective space Pm.

The way to obtain this finite map is almost algorithmic:

Input: X ⊆ Pn closed.

• If X 6= Pn, choose some p ∈ Pn \X.

• Take the image of X under the stereographic projection πp(X).

• The projection gives us a regular map X −→ πp(X), so by Theorem 3.20 πp(X) ⊆ Pn−1 is closed.

• By Theorem 3.33 πp : X −→ πp(X) is a finite map.

• If πp(X) ( Pn−1, then repeat. The finite map will be the composition of the projections.

The same works if you replace projective with affine.

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3.8 Rational Maps

Definition 3.37. {f/g : f, ghomogeneous of same degree , g /∈ I(X)}

Lemma: k(X) ' k(Ui ∩X).

Let X be an irreducible quasi-projective variety. Recall that for an irreducible quasi-projective variety X ⊆ Pnwe write OX = {f = g

h : deg g = deg h and h /∈ IP(X)}. If MX = {f = gh ∈ OX : g ∈ IP(X)}, then

k(x) = OX/MX is the function field of X. If U ⊆open X, then, since g vanishes on X if and only if it vanisheson U , then k(X) = k(U). In particular, k(X) = k(X), where X ⊆ Pn is the projective closure. So when talkingabout function fields, we may restrict to affine or projective varieties if we want to.A function f ∈ k(X) is regular at a point P ∈ X if we can write it as f = g

h with h(P ) 6= 0. Then f(P ) = g(P )h(P )

is the value of f at P . The domain of definition of f is

dom(f) = {P ∈ X : f is regular at P}.

A rational mapf : X Pm

on X is given by f(P ) = (f0(P ) : · · · : fm(P )) for rational functions fi ∈ k(X). The map f is regular at P ifP ∈

⋂dom(fi) and (f0(P ), . . . , fm(P )) 6= (0, . . . , 0). The domain of regularity of f , dom(f), is defined as

expected. It is an open set U ⊆ X. So a rational map on X is a regular map on an open subset of X.If Y ⊆ Pm is a quasi-projective variety and f : X 99K Pm, then we say that f maps X to Y if there exists anopen set U ⊆ dom(f) ⊆ X such that f(U) ⊆ Y . The union of all such U is U , the domain of definition off . The set im(f) := f(U) ⊆ Y is the image of X in Y . As in the affine case, if im(f) is dense in Y , then fdefines an inclusion of fields f∗ : k(Y ) ↪→ k(X). Compositions are again defined only if the images are dense. Ifa rational map has a rational inverse, then f is birational or a birational equivalence. We say in this casethat X and Y are birational. In this case the fields k(X) and k(Y ) are isomorphic.

Proposition 3.38. Two irreducible quasi-projective varieties X and Y are birational if and only if they containisomorphic open subsets U ⊆ X and V ⊆ Y .

Proof. Let f :99K Y and g = f−1 : Y 99K X. Let U1 and V1 be the domains of definition of f and grespectively. By assumption f(U1) is dense in Y so f−1(V1) ∩ U1 is open and nonempty. Similarly for g. Weset U := f−1(V1) ∩ U1 and V := g−1(U1) ∩ V1. It is easy to check that f(U) = V and g(V ) = U , and fg = 1and gf = 1.

As o consequence, we have that the Grassmannian is a rational variety: as it contains a dense open subsetisomorphic to Ak(n−k), it is birational to Pk(n−k).

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Chapter 4

Dimension

Dimension is probably the geometric concept that has to agree most with our intuition. A point has to havedimension zero, and An and Pn have to have dimension n. Putting together finitely many objects of thesame dimension has to have the same dimension. This brings us closer to making the concept more precise:Theorem 3.36 could help. If f : X −→ Am (or f : X −→ Pm) is a finite map, then the dimension of X shouldbe m. Dimension should also be local, so it makes sense to start by looking at irreducible affine varieties.The first question that arises is, could there be two different finite maps f : X −→ Am and g : X −→ An withm 6= n? Suppose X is irreducible, so we can talk about K(X), the function field of X (cf. Section 3.2). Thenf∗ : K(An) = K(x1, . . . , xn) −→ K(X) is injective, so K(X) is an algebraic filed extension of K(x1, . . . , xn)(algebraic because the map is finite). This inspires the following definition for irreducible quasi-projectivevarieties.

Definition 4.1. The dimension of an irreducible quasi-projective variety X is the transcendence degree overK of its function field K(X). We denote this by

dimX := tr.degKK(X).

The dimension of a reducible quasi-projective variety X is

dimX := max{dimXi : Xi irreducible component of X}.

If Y ⊆closed X, then the codimension of Y in X is codimX Y := dimX − dimY .

An affine/projective curve is an affine/projective variety of dimension one. An affine/projective surface is anaffine/projective variety of dimension two. Varieties of dimension n ≥ 3 are called n-folds.

Remark 4.2. a. If X and Y are birationally equivalent, then dimX = dimY .

b. If U ⊆open X, then dimU = dimX. In particular, when studying dimension, we may often reduce to theaffine case.

Examples. 1. The dimension of affine and projective space is what one expects: dimAn = dimPn = n. Inparticular, if n 6= m, then An and Am are not birational.

2. If X is a single point, then K(X) = K, thus X has dimension zero. The same holds thus for finite sets ofpoints. Conversely, if dimX = 0, then X is a finite set. It is enough to assume that X ⊆ An is irreducible.Then the coordinate functions x1, . . . , xn ∈ A(X) are algebraic over K, that is they satisfy a polynomialequation. Hence each can take only finitely many values.

3. If X and Y are irreducible varieties, then dim(X × Y ) = dimX + dimY. (see [Sha13, p.67, Ex.1.33] for aproof).

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4. The Grassmannian Gr(k, n) has a finite open cover with Ak(n−k), so it has dimension k(n− k).

Theorem 4.3. Let X ⊆ Y be quasi-projective varieties.

(i) dimX ≤ dimY .

(ii) If Y is irreducible and X ⊆closed Y with dimX = dimY , then X = Y .

Proof. [Sha13, p.68].

Theorem 4.4. Every irreducible component of a hypersurface in An or Pn has codimension one.

Proof. • It is enough to consider the affine case: X = Z(f) ⊆ An, with f ∈ K[x1, . . . , xn].

• Factorize f = f1 · · · fr into irreducible factors. Then X = X1 ∪ · · · ∪Xr with Xi = Z(fi). It is enough toprove this for Xi.

• Xi is irreducible. (small proof?)

• Assume xn actually appears in fi. We claim that x1, . . . , xn−1 ∈ A(Xi) are algebraically independent. Ifnot, then there exists a polynomial relation g(x1, . . . , xn−1) = 0 ∈ A(X), which means fi|gk for some k.But this is impossible, since xn does not appear on the right.

• So dimXi ≥ n− 1. Since X 6= An, we conclude by Theorem 4.3.

Theorem 4.5. Let X ⊆ An be a closed subset with all components of dimension n−1. Then X is a hypersurface(i.e. the zero locus of just one polynomial), and I(X) is principal.

Proof. • Assume X is irreducible.

• dimX = n− 1⇒ X 6= An ⇒ ∃f ∈ I(X) ⊆ K[x1, . . . , xn].

• Since X is irreducible, we may also assume F is (otherwise replace it by an irreducible factor).

• So X ⊆ Z(f), with the latter irreducible and of dimension n− 1. Conclude X = Z(f) by Theorem 4.3.

• Clearly I(X) = (f) by the irreducibility of f .

Remark 4.6. The above theorem does not hold for pure codimension one subvarieties of varieties in general(see Corollary 4.16)

One can prove completely analogously that if X ⊆ Pn1 × · · · × Pnk is a closed subset whose components havedimension n1 + · · ·+ nk − 1, then X is defined by one multihomogeneous equation.

4.1 Intersections with Hypersurfaces

We saw that hypersurfaces, that is zero loci of one polynomial, are the same thing as codimension one closedsubsets. This is cannot be generalized to higher codimension (see Exercise 4 on Sheet 8). We will see that thecase when this generalization holds (i.e. codimension = minimal number of generators of the ideal/ or minimalnumber of equations defining the close set) is particularly well behaved, and in many senses easier to handlethan the general situation.We say that a closed variety X ⊆ Pn with codimPn X = r is a complete intersection if I(X) = (f1, . . . , fr)for some f1, . . . , fr ∈ S, and that X is a set theoretical complete intersection if there exists f1, . . . , fr suchthat X = Z(f1, . . . , fr). In the latter, the ideal generated by f1, . . . , fr need not be radical.

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In trying to understand this phenomenon, one is immediately confronted with the question: how does the di-mension of the zero locus X change if one more equation is added. That is, what happens when we intersect Xwith a hypersurface.

Let X ⊆ Pn be a projective variety (i.e. a closed subset). For a form F ∈ S = K[x0, . . . , xn] \ I(X) writeXF = X ∩ Z(F ) for the closed subvariety defined by F on X. Let us first see that there always exist forms ofany degree F /∈ I(X).

Lemma 4.7. For any closed projective set X ⊆ Pn and for any d ∈ N there exists a form F ∈ Sd which doesnot vanish on any component of X.

Proof. • Let X = X1 ∪ · · · ∪Xr be the decomposition of X into irreducible components.

• Choose pi ∈ Xi for every i = 1, . . . , r.

• As vanishing for a form at a point is independent of multiplication with a nonzero scalar, we can viewforms of degree d as points in P(Sd). The forms vanishing at pi form a proper closed subset of P(Sd), thusthe forms vanishing at any of the (finitely many) points pi also form a closed subset. Just pick some F inthe complement.

Theorem 4.8. If X is a projective variety and F a form not vanishing on any irreducible component of X,then dimXF = dimX − 1.

Proof. • Set X0 := X.

• By Lemma 4.7 choose F0 which does not vanish on any component of X0.

• By Theorem 4.3, we have that the dimension of each component of X0 drops when intersecting withZ(F0). So also dim(X0 ∩ Z(F0)) < dimX0.

• Set X1 := X0 ∩ Z(F0) and repeat the procedure for some F1 not vanishing on any component of X1, ofthe same degree as F0. (Notice that X1 may have more irreducible components than X0.)

• We have now a chain of algebraic sets of strictly decreasing dimension

X = X0 ) X1 ) . . .

and a sequence of forms F0, F1, . . . of the same degree such that Xi+1 = Xi ∩Z(Fi) = X ∩Z(F0, . . . , Fi).

• If dimX = d, we must have dimXd ≤ 0, and thus Xd+1 = ∅. In particular, this means that the formsF0, . . . , Fd have no common zero on X.

• We can define the map ϕ : X −→ Pd by

ϕ(p) := (F0(p) : · · · : F0(p)),

which by Theorem 3.34 is finite when restricted to ϕ : X −→ ϕ(X).

• We know by definition that finite maps preserve dimension, so dimX = dimϕ(X) = d.

• As ϕ(X) ⊆ Pd is closed (Corollary 3.29) and by the previous point of the same dimension it follows fromTheorem 4.3 that ϕ(X) = Pd.

• Assume now by contradiction that dimX1 < d − 1. Then we must have already that Xd = X ∩Z(F0, . . . , Fd−1) = ∅. In particular there is no point p ∈ X such that ϕ(p) = (0 : · · · : 0 : 1) – acontradiction.

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Corollary 4.9. A projective variety contains subvarieties of any dimension 0 ≤ s < dimX.

A useful consequence of Theorem 4.8 is a recursive definition of dimension:

Corollary 4.10. If X is an irreducible projective variety then

dimX = 1 + sup{dimY : Y ( X is a proper subvariety}.

For a not necessarily irreducible projective variety X we have that the dimension of X is the maximal integerd for which there exists a strictly decreasing chain

X ⊇ Y0 ) Y1 ) · · · ) Yd ) ∅,

with Yi ⊆ X irreducible subvarieties.The consequences mentioned so far were immediate. The next one needs a bit of proof.

Corollary 4.11. If X ⊆ Pn is a projective variety, then

dimX = n− s− 1,

with s = max{dimL : L ⊆ Pn is a linear subspace with L ∩ X = ∅}. Notice that this is not cheating, asdimension was defined independently for linear subspaces.

Proof. Let L ⊆ Pn be a linear subspace of dimension s. We first show that dimX ≤ n − s − 1. Assume thecontrary, that is s ≥ n− dimX. Then L can be defined by at most n linear equations. Applying Theorem 4.8successively we get that dim(X ∩ L) ≥ 0, and thus X ∩ L 6= ∅. For the other inequality we just use the samemethod as the proof of Theorem 4.8 for forms of degree 1.

Another obvious but powerful consequence is the following.

Corollary 4.12. Let F1, . . . , Fr ∈ S be forms on an n-dimensional projective variety X. Then dimZX(F1, . . . , Fr) ≥n− r.

While the next result is still a direct consequence, its importance makes it deserve to be called a theorem.

Theorem 4.13 (Existence of Common Zeros). Let X be an n-dimensional projective variety, and r ∈ N withr ≤ n. Then any r forms on X have a common zero on X. In particular, n homogeneous equations in n + 1variables always have a nonzero solution.

Corollary 4.14. Any two curves in P2 intersect.

Example 4.15. This is not in general true for curves on a surface. We have seen that P1 × P1 ' Q =Z(x0x3 − x1x2) ⊆ P3 contains families of disjoint lines. In particular, this implies that, while P2 and Q arebirational, they are not isomorphic. (They are birational because every quadric in Pn is rational).

Corollary 4.16. There exist curves C on surfaces Q ⊆ P3 that cannot be defined by setting to zero a singleform on Q. In particular Theorem 4.4 fails already for surfaces.

Proof. We know that Q = Z(g) from Example 4.15 contains disjoint curves C1 and C2. If they were eachdefined by one equation on Q: f1 and f2 respectively, then Z(g, f1, f2) 6= ∅ by Corollary 4.12.

‘

Theorem 4.17. Let X be an irreducible projective variety and F a form not vanishing on X. Every componentof X ∩ Z(F ) has dimension dimX − 1.

Proof. The proof is relatively long. Check [Sha13, Theorem 1.23, p.73].

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Corollary 4.18. Let X ⊆ PN be an irreducible quasi-projective variety and F a form that is not identicallyzero on X. Every nonempty component of X ∩ Z(F ) has codimension one.

Proof. • X ⊆open X ⊆closed PN .

• X is irreducible ⇒ X is irreducible ⇒ dimX = dimX.

• Theorem 4.17 ⇒ X ∩ Z(F ) =⋃Yi with dimYi = dimX − 1.

• X ∩ Z(F ) = (X ∩ Z(F )) ∩X =⋂

(Yi ∩X),

• Yi ∩X is open in Yi, and if it is not empty, then dim(Yi ∩X) = dimX − 1.

4.2 Lower Bounds on Dimension

Corollary 4.19. Let X ⊆ Pn be an irreducible d-dimensional quasi-projective variety, and Y = ZX(F1, . . . , Fr).Then every (nonempty) component of Y has dimension ≥ d− r.

Theorem 4.20. Let X,Y ⊆ Pn be irreducible quasi-projective varieties of dimensions d and e respectively.Then any irreducible component1 Z of X ∩ Y has dimZ ≥ d+ e− n.Moreover, when X and Y are projective, then d+ e ≥ n implies X ∩ Y 6= ∅.

Proof. • For the first part, it is enough to assume that X,Y ⊆ An are affine varieties.

• Let ∆ = {(x, x) : x ∈ An} ⊆ An × An = A2n.

• X ∩ Y ' (X × Y ) ∩∆ = (X × Y ) ∩ Z(x1 − xn+1, . . . , xn − x2n)

• Conclude by Example 4 (dimX × Y ) and Corollary 4.19.

• For the second part, take C(X), C(Y ) ∈ An+1 the affine cones over X and Y respectively.

• From Exercise Sheet 8 we have dimC(X) = dimX + 1 and dimC(Y ) = dimY + 1.

• Their intersection is not empty (the cone point is in it), and by the first part it has components ofdimension ≥ d+ 1 + e+ 1− n+ 1 = d+ e− n+ 1. This the cone over the intersection.

Another way to phrase the previous theorem is

codimX

m⋂i=1

Yi ≤m∑i=1

codimX Yi (4.1)

where Yi are subvarieties of X.1There may be none if the intersection is empty.

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4.3 Dimension Of Fibres

The fibre of a morphism f : X −→ Y over a point q ∈ Y is f−1(q). As morphisms between quasi-projectivevarieties (i.e. regular maps) are continuous in the Zariski topology, fibers are closed subvarieties of X. We alsohave X =

⊔q∈Y f

−1(q).

Theorem 4.21. Let X and Y be irreducible quasi-projective varieties and let f : X −→ Y be a surjectiveregular map. Then the following hold.

(i) dimX ≥ dimY .

(ii) dimZ ≥ dimX − dimY for any q ∈ Y and any irreducible component Z of the fibre f−1(q).

(iii) There exists ∅ 6= U ⊆open Y such that

dimf−1(q) = dimX − dimY, ∀ q ∈ U.

Proof. Clearly the problem is local, i.e. we may replace Y with an open neighborhood U of q and X by theopen subset f−1U . So we assume that Y is affine: Y ⊆ An. Denote by d := dimX and e := dimY .

(i) Follows from part (ii)

(ii) Let Y0 ) Y1 ) · · · ) Ye be the chain of (irreducible) subvarieties as in the proof of Theorem 4.8. That is

Yi = Y ∩ Z(g1, . . . , gi).

In particular, Ye has dimension zero and is thus a finite set. We may even assume that {q} = Ye, that isthe only point of Y at which all gi vanish is q. This can be achieved by a cunning choice of U (above).So {q} = ZY (g1, . . . , ge), which means that

f−1(q) = ZX(f∗(g1), . . . , f∗(ge)).

We conclude by Corollary 4.19.

(iii) It is here where the irreducibility starts playing a role. This allows the replacement of X and Y with opensubsets which, due to irreducibility, will be dense. Then, the definition of dimension as transcendencedegree of A(X) and A(Y ) over K, toghether with the ring inclusion f∗ : A(Y ) ↪→ A(X) lead to a proofusing algebraic dependence. See [Sha13, p.76, proof of Thm.1.25].

Part (ii) above holds for all points in Y , that is all fibres have the same dimension, when the morphism is flat.This will be seen later on, in the more general setup of schemes. For the moment here are some examples wherethe dimension of the fibres jumps.

Example 4.22. 1. f : A2 −→ A2, with f(x, y) = (x, xy). This is not surjective, but you may choose somerestriction...

2. Let X = Z(xy − x) ⊆ A2 and f = π1|X the restriction of the projection π1 : A2 −→ A1, (a, b) 7−→ a.

Corollary 4.23. Let f : X −→ Y be a regular map and k ∈ N. The following sets are closed in Y :

Yk := {q ∈ Y : dim f−1(q) ≥ k}.

Proof. Let d = dimX and e = dimY . By Theorem 4.21 Part (i) we have Yd−e = Y . By Part (ii) there existsa closed subset Y ′ ( Y such that Yk ⊆ Y ′ if k > d − e. Now decompose Y ′ = Z1 ∪ · · · ∪ Zr into irreduciblecomponents, and look at the restrictions fi : f−1(Zi) −→ Zi. Since Zi ⊆ Y ′ ( Y , we have dimZi < dimY forevery i, and we can conclude by induction on the dimension.

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4.4 Criterion for Irreducibility

Theorem 4.24. Let X and Y be projective varieties and f : X −→ Y a surjective regular map. If Y isirreducible and for all q ∈ Y the fibres f−1(q) are irreducible and of the same dimension, then X is irreducible.

Proof. • X = X1 ∪ · · · ∪Xr irreducible decomposition.

• f(Xi) ⊆closed Y for each i.

• Y is irreducible and Y = f(X1) ∪ · · · ∪ f(Xr) then ∃ i such that f(Xi) = Y .

• All fibers have the same dimension. Call it n := dim f−1(y).

• ∀ i with f(Xi) = Y , by Theorem 4.21 (iii), ∃ ni ∈ N and Ui ⊆open Y such that dim f−1(q) = ni for allq ∈ Ui.

• ∀ i with f(Xi) 6= Y , set Ui := Y \ f(Xi).

• Let q ∈⋂Ui (this is not empty because Y is irreducible). Since f−1(q) is irreducible, we have f−1(q) ⊆ Xi

for some i. Assume harmlessly that i = 0.

• Let f0 = f |X0: X0 −→ Y . So f−1(q) = f−1

0 (q) (for the chose special q) which implies n = n0.

• So the dimension of the fibres of the restriction to an irreducible component is the same as the originaldimension of the fibres.

• Let us look at all the fibres now. Since f0 is surjective we have

∅ 6= f−10 (q) ⊆ f−1(q), ∀ q ∈ Y.

By Theorem 4.21 (ii) dim f−10 (q) ≥ n0 = n, so, as both are closed and f−1(y) is irreducible, we have

f−10 (q) = f−1(q) for all q ∈ Y , and thus X0 = X.

We get a consequence which we had already proven directly: the product of irreducible varieties is irreducible.

Reading assignment: [Sha13, Chapter 1, Section 6.4 (p.77-79)] It is a good recap of the things we did so far,and it forms a good basis for future results.

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Chapter 5

Singularities

We are now aiming at defining singular points. This is a local property, so we may restrict by Lemma 3.4 toaffine varieties.

5.1 The Local Ring

Let X ⊆ An be an affine variety and p ∈ X be a point. The local ring at the point p is OX,p := A(X)mp . Wesometimes write simply Op. This can also be defined by regular functions f

g in some neighborhood U = D(g)

of p. In other words, we may view Op as the ring of germs of regular functions on X near p. We say fg = f ′

g′

if there exists h ∈ A(X) \ mp such that h(fg′ − f ′g) = 0 in A(X). In particular, this means that fg and f ′

g′

coincide on the open neighborhood D(hgg′).One can localize at any irreducible subvariety Y , by localizing at the corresponding prime. This produces thefunction field: K(Y ) = OY /mY .Every local ring is Noetherian.

5.2 The Tangent Space

The tangent space to an affine variety X ⊆ An at a point p is the set of all lines through p tangent to X. Tomake this precise, we start by choosing coordinates on X such that p = (0, . . . , 0) = 0 (we just call it 0). Thenevery line containing 0 has the form

L = {tq | t ∈ K}

for some fixed point 0 6= q ∈ An. Fix such a line and choose a set of generators

I(X) = 〈f1, . . . , fm〉

of the ideal of X. Then the intersection of L with X is given by

X ∩ L = {tq : f1(tq) = · · · = fm(tq) = 0}.

This reduces our quest to solving polynomial equations in one variable t. As K[t] is a principal ideal domain,there exists a gcd(f1(tq), . . . , fm(tq)) = f(t). As K = K this polynomial factors into linear forms:

f(t) = c∏

(t− αi)ki , with αi 6= αj , when i 6= j.

So ki is the multiplicity of the root αi, which is naturally interpreted as the multiplicity of intersection of Land X in the point αiq. We have chosen the coordinates such that 0 ∈ X and we chose L 3 0, so what we

39

care is the intersection multiplicity of the line L and X at the point 0, which is the multiplicity of the rootαi = 0 of f . If all fi(tq) are the zero polynomial, then we say that the intersection multiplicity is +∞. In otherwords, the intersection multiplicity is the highest power of t which divides every fi. This must be ≥ 1, since0 ∈ L∩X. What happens if we choose other generators for I(X)? Nothing changes, because, as K[t] is a PID,the polynomial f is actually the generator of the ideal 〈f(tq) : f ∈ I(X)〉, that is

f(t) = gcd(f(tq) : f ∈ I(X)).

Finally, here is the definition we were building up to

Definition 5.1. A line L is tangent to X at the point p ∈ X if the intersection multiplicity of X and L at pis at least 2.

Now back to p = 0, and unfix the line Lq = {tq : t ∈ K}, that is we let q vary, and study the conditionsfor Lq to be tangent to X. First, since 0 ∈ X, all constant terms of fi must be zero. We write now `i for thedegree one part of fi, that is

fi = `i + gi,

where every gi has only monomials of degree higher than two. This means that fi(tq) = `(q)t+ g′i(tq)t2. Thismeans that, the multiplicity of the root 0 of fi(tq) is higher than one, if and only if `(q) = 0. This means that

Lq is tangent to X ⇐⇒ q ∈ Z(`1, . . . , `m).

Clearly the whole discussion can be modified slightly so that it describes conditions of tangency at any pointp ∈ X. Nothing more illuminating happens. Just notation becomes heavier.

Definition 5.2. The geometric locus of points belonging to lines tangent to X at p is called the tangentspace to X at p. It is denoted by ΘX,p or just Θp.

This means thatΘX,p =

⋃Lq, with Lq tangent to X at p.

In particular, when p = 0, with the notation before the theorem we have

ΘX,0 = Z(`1, . . . , `m), (5.1)

which shows that the tangent space is a linear subspace of An. This is the pleasing for our intuition of tangentspace to a geometric object, but in order to reveal the intrinsic nature of the tangent space we need to fix avector space structure on this affine subspace. This means, we have to choose and fix an origin. Now there isone very obvious and natural choice for this: p, the only point we can actually name in the linear space Θp.For the moment we will not care much about this choice, but this will become crucial once we look at duals.

Examples. 1. The tangent space to An at any point is just An itself. The same holds for any linear space:ΘL,p = L.

2. If X ⊆ An is a hypersurface, with I(X) = 〈f〉 with 0 ∈ X and f = ` + g, then Θ0 = Z(`). So, if ` 6= 0,then dim Θ0 = n− 1. However, if ` = 0, then we get Theta0 = An.

3. Take the parabola X = Z(y−x2). The tangent to it at 0 is the x-axis: Z(y). If we add a linear irreduciblecomponent to X, say Y = X ∪Z(y), that is Y = Z(y2− yx2), then, even if the tangent space at 0 to eachcomponent is the line Z(y), the tangent space to their union is the whole plane A2.

Remark 5.3. The equations of the tangent space are to be read so directly from the linear part of the definingequations only when we consider the tangent space at 0. In general the equations are obtained in analogy to:

n∑i=1

∂f1

∂xi(0)xi = · · · =

n∑i=1

∂fm∂xi

(0)xi = 0.

40

We will expand on this last remark now. Let us first recall some partial derivatives and differentials. For apolynomial with coefficients in an arbitrary field partial derivatives with respect to one of the variables stillmake sense, even if limits may not be properly defined. This is done formally by simply setting

∂(c · xa11 . . . xann )

∂xi= aic · xa11 . . . xai−1

i . . . xann .

and applying the general rules of derivation for sums and products. As Hartshorne puts it,“funny thingscan happen in charateristic p > 0”, because ∂xp

∂x = pxp−1 = 0xp−1 = 0. Anyway, the partial derivative ofa polynomial is always a polynomial. The differential of a polynomial f ∈ K[x1, . . . , xn] at a point p =(p1, . . . , pn) ∈ An is defined as

dpf =

n∑i=1

∂f

∂xi(p)(xi − pi).

From the rules of derivation which hold for ∂f∂xi

we get also

dp(f + g) = dpf + dpg,

dp(fg) = f(p)difpg + g(p) dpf.

Here is an explicit example

d(1,√

2)(y2 − x3 − x2) = [(−3x2 − 2x)(1,

√2)](x− 1) + [(2y)(1,

√2)](y − 2)

= −5x+ 5 + 2√

2y − 4√

2

= d(1,√

2) y2 + d(1,

√2)(x

2(x+ 1))

This can be generalized to higher orders to obtain the Taylor expansion

f(x) = f(p) + f (1)(x) + · · ·+ f (d)(x)

with f (1) = dpf , and, in general, f (i) a polynomial of degree i homogeneous in the variables (xj − pj) (not inthe xj !).Coming back to the tangent space, the equations given in (5.1) take for an arbitrary point p ∈ X, withI(X) = (f1, . . . , fm) the form

ΘX,p = Z(dpf1, . . . ,dpfm). (5.2)

5.3 Intrinsic Definition of the Tangent Space

Definition 5.2 was given in terms of the equations defining X as a subset of An. What if we have an isomorphismX −→ Y ⊆ Am? The corresponding tangent spaces should also be isomorphic. A solution to this problem isgiven by Zariski in a 1947 paper, in which the tangent space was described intrinsically. Intrinsically means inour setup that the tangent space to a point is expressed only in terms of the ring of rational functions whichare regular at the given point. This local ring was recalled at the beginning of the current chapter. The generalstrategy of viewing algebraic varieties independently of an embedding is a central idea behind the developmentof algebraic geometry beginning with the second half of the XX century.In the last part of Section 5.2 we described the equations of the tangent space in terms of differentials ofpolynomials. We now want to define differentials of polynomial functions on X, that is of elements in A(X) =K[x]/I(X). We need to choose a representative for this, and then show that what we care about is independentof this. What we care about is defining a linear function on the tangent space, just as the differential of apolynomial is a linear function on the whole affine space.

41

Let g, h ∈ K[x] be two representatives of the same regular function on X. That is [g] = [h] ∈ A(X), orequivalently g − h ∈ I(X) = (f1, . . . , fm). This means there exist polynomials u1, . . . , um ∈ K[x] such thatg − h =

∑uifi. Chose p ∈ X and regard the difference dpg − dph. By the additivity of derivation we get

dpg − dph = dp(g − h)

= dp(u1f1 + · · ·+ umfm)

=∑ri=1

(fi(p) dpui + ui(p) dpfi

)=

∑ri=1 ui(p) dpfi,

where the last step is due to p ∈ X = Z(f1, . . . , fm). By (5.2) we thus have that on the tangent space ΘX,p

the difference is zero. This means we can define for any regular function a linear form on the tangent space.

Definition 5.4. Let X be an affine variety, p ∈ X be a point, and [g] ∈ A(X) a regular function on X. Thedifferential of [g] at the point p is the linear form dp[g] : ΘX,p −→ K given by

dp[g] := dpg|Θp .

The right hand side above denotes the restriction of the differential of the polynomial g to the tangent space.By the above discussion this is independent of the polynomial g representing the function. When we are lessinterested in the representative, and just have ϕ ∈ A(X), we simply write dpϕ.

In Definition 5.2 Θp was introduced as a linear subspace of an affine space. This means that after choosing apoint to play the role of the zero vector, we can view Θp as a K-vector space as well. This will be done from nowon, and Θ∗p will denote the dual vector space, that is the K-vector space of linear forms on Θp. The connectionwith the local ring will be given in terms of this dual space.The properties of differentiation for sums and products clearly still hold. Given that the differential of a constantfunction is the zero function, we get

dp(λf + µg) = λ dpf + µdp g, ∀ λ, µ ∈ K and ∀ f, g ∈ A(X).

This means that dp : A(X) −→ Θ∗p is a K-vector space homomorphism. Again because dpc = 0 for all constantsc ∈ A(X) we may restrict dp to the ideal (and thus also K-subspace) mp = {f ∈ A(X) : f(p) = 0}. Thus Wecan now state the result which gives the intrinsic nature of the tangent space.

Theorem 5.5. The K-linear map dp : mp −→ Θp defined above induces an isomorphism

mp/m2

p' Θ∗p.

Proof. In other words, the map dp : mp −→ Θp is surjective with kernel m2p.

Surjectivity A linear form of ϕ : Θp −→ K is a linear function which maps p to zero. In particular, it isinduced by a linear element from f ∈ mp = 〈x1 − p1, . . . , xn − pn〉, that is f = λ1(x1 − p1) + · · ·+ λn(xn − pn)and ϕ(q) = f(q) for all q ∈ Θp. For each such f we have dp f = f, so we conclude surjectivity.Kernel If dp g = 0 ∈ Θ∗p with g ∈ mp, this means

dp g = λ1 dp f1 + · · ·+ λm dp fm,

where I(X) = 〈f1, . . . , fm〉. As p ∈ X, we have fi ∈ mp for i = 1, . . . ,m. By definition we are allowed toreplace g with any g + f for any f ∈ I(X), without altering dp g. So set

g1 = g − λ1f1 − · · · − λmfm.

Now g1 has no part of degree 0 or 1, so it must belong to m2p (as an ideal of K[x1, . . . , xn] for first). But this

means that g, as an element of A(X), belongs to the ideal mp2 ⊆ A(X). Bam!

42

OK, this (i.e. the degree 0 or 1 part) is not literally true, unless p = (0, . . . , 0). On the one hand, we mayassume that p = 0. On the other hand, even without this assumption, not having a degree zero part means justg1 ∈ mp. The degree one part corresponds to the part c1(x1 − p1) + · · ·+ cn(xn − pn) of g which has to cancelout with the corresponding one of λ1f1 − · · · − λmfm, which also belongs to mp.

Using that for finite dimensional K-vector space V we have a canonical isomorphism (V ∗)∗, where V ∗ =HomK(V,K), we get the following consequence.

Corollary 5.6. Let p be a point of an affine variety X, and mp ⊆ A(X) the maximal ideal of regular functionson X vanishing at p. Then the tangent space to X at p is isomorphic (as a K-vectors space) to the dual spaceof mp

/m2

p.

The cotangent space to X at p is mp/m2

p.

Before we accept this intrinsic definition, let us make sure that it is truly invariant under isomorphisms. So letf : X −→ Y be any (i.e. not necessarily an isomorphism) regular map between affine varieties. And let p ∈ Xand q := f(p) ∈ T . Then f∗ : A(Y ) −→ A(X) with

f∗(mq) = f∗({g ∈ A(Y ) : g(q) = 0}

)=

{f∗(g) ∈ A(X) : g(f(p)) = 0

}=

{f∗(g) ∈ A(X) : f∗(g)(p) = 0

}⊆

{h ∈ A(X) : h(p) = 0

}= mp.

This implies f∗(m2q) ⊆ m2

p in the obvious way, and thus f induces a linear map

f∗ : mq/m2

q−→ mp

/m2

p.

The notation ∗ is used both for the pullback and for the dual vector space. This is not a coincidence, the two arerelated in spirit. That is, for a vector space homomorphism ϕ : V −→ W ,we have ϕ∗ : W ∗ −→ V ∗. Applyingthis to f∗ mq

/m2

q−→ mp

/m2

p, we get a map “

(f∗)∗” which we will call the differential of f , and denote by

dp : ΘX,p −→ ΘY,y.

Next, we are going to make the following statement more precise: The tangent space is a local invariant.

Theorem 5.7. Let p be a point of a variety X, and let (Op,mp) denote the local ring of p. The tangent spacesatisfies

ΘX,p ' mp/m2

p.

Proof. The proof works in the same spirit as the proof of Theorem 5.5. The only difference is that we have toconsider rational functions f

/g , with g(p) 6= 0. The differential of such a function is given by

dp

( f/g)

=g(p) dp f − f(p) dp g

g2(p).

The intrinsic definition of the tangent space at p ∈ X is

Θp =( mp

/m2

p

)∗,

where mp is the maximal ideal of the local ring Op. By Theorem 5.7 this is also the tangent space at p to anyaffine neighbourhood of p.

Remark 5.8. If we have p ∈ X with dimK ΘX,p = m, then X is not isomorphic to any subvariety of an affinespace An with n < m.

For any quasi-projective variety we define the tangent space locally, as above.

43

5.4 Singular Points

As we are interested in a local property, we may assume that X is an affine variety.

5.4.1 On an irreducible variety X

Let X ⊆ An be a closed irreducible set, with I(X) = (f1, . . . , fm), and consider the tangent bundle

ΘX := {(p,q) : p ∈ X and q ∈ ΘX,p} ⊆ X × An.

This is a closed subset of X×An given by the equations (5.2). Consider the first projection π1 : X×An −→ X,restricted to ΘX , so π1 : ΘX −→ X is a regular map between closed affine varieties , which is also clearlysurjective. The fibres are (canonically isomorphic to) the tangent space at the given point π1(p)−1 = ΘX,p ⊆ An,(seen as an affine space, but we will only care about the dimension anyway). By Theorem 4.21 there exists ans ∈ N such that dim Θp ≥ s for all p ∈ X, and by Theorem 4.21 (iii) and Corollary 4.23 the set

Sing(X) = {p ∈ X : dim Θp > s} ( X

is a proper closed subvariety of X. So, as X is irreducible, it is a subvariety of strictly smaller dimension.

Definition 5.9. For an irreducible variety X let

s := minp∈X

dim Θp.

A point p ∈ X is a nonsingular point (or a smooth point if dim Θp = s, and p is a singular point ifdim Θp > s 1. We say that X is a nonsingular variety if X is nonsingular at every point.

This definition does not seem very practical, because it implies finding this number s at first. The next theoremtakes care of this.

Theorem 5.10. Let X be an irreducible variety and p a point on X. Then p is a smooth point if and only ifdim Θp = dimX.

Proof. Our strategy is to reduce this to hypersurfaces. The statement of the theorem is essentially that dim Θp ≥dimX for all p, and that equality holds on a nonempty open subset of X. In other words, that the s inDefinition 5.9 is dimX.By Theorem 2.37, every irreducible closed set is birational to a hypersurface Y , which is embedded in some otheraffine space Am. Let ϕ : X Y be such a birational map. By Proposition 3.38, there exist nonempty

open sets U ⊆ X and V ⊆ Y such that the restriction of ϕ is an isomorphism: ϕ : U∼−→ V .

Claim: For a hypersurface Y , we have dim Θq = dimY on a nonempty open subset W ⊆ Y .

Using the Claim and the fact that Y is irreducible (because it is isomorphic to X), we have ∅ 6= V ∩W ⊆open Y .Thus also ∅ 6= ϕ−1(W ∩ V ) ⊆open U ⊆open X. Since the dimension of the tangent space is local and invariantunder isomorphism, we get dim Θp = dimY = dimX for all p ∈ ϕ−1(W ∩ V ). So it only remains to prove theclaim.

Proof of Claim: Let Y ⊆ Am be our hypersurface, with I(Y ) = 〈f〉, where f ∈ K[y1, . . . , ym]. The equation ofΘY,q is then

ΘY,q = Z(∑ ∂f

∂yi(q)(yi − qi)

).

This linear space has dimension dimY if only if the linear form defining it is not trivial. This linear form isnontrivial on an open set, if and only if the partial derivatives ∂f

∂yiare not all identically zero on Y . If char K = 0,

this would mean that f is constant – a contradiction. If char K = p > 0, this means that all variables appearin f with exponents that are multiples of p. But, by the Frobenius map, this means that f = fp1 for somef1 ∈ K[x], which contradicts I(Y ) = 〈f〉.

1We also say that X is nonsingular, respectively smooth, at p.

44

5.4.2 On any variety

WhenX is a quasi-projective variety, a point p ∈ X is nonsingular if it is nonsingular on an affine neighbourhoodU of p, which is equivalent to dim ΘX,p = dimpX. We will see later (Theorem 5.21) that smooth points cannotbelong to more than one irreducible component. A quasi-projective variety is nonsingular, or smooth, if everypoint is nonsingular.

5.5 The Tangent Cone

The dimension of the tangent space is usually not enough to distinguish between singularities. This is especiallyclear for curves in the plane, for which dim Θp = 2 for all singular points. Check out the four singularities inExercise 5.1 Hartshorne (page 35-36). The tangent cone helps to distinguish there.

Let X be an irreducible affine variety. The tangent cone to X at p ∈ X is an analogue of limiting positions ofsecants through that point. An illuminating example are the tangent lines intersecting the cusp ZP(y2z − x3),respectively the nodal curve ZP(y2z − x2z − x3), only at the origin. This intersection criterion is not worthpursuing, as it cannot be generalized properly.Let X ⊆ An be our irreducible affine variety, and assume p = (0, . . . , 0). This assumption is frequent whenstudying singularities/tangent spaces. One advantage is that the distinction between the tangent space as anaffine space and a vector space becomes blurrier. Define now

X := {(a, t) ∈ An × A1 : ta ∈ X}.

If we view K[x1, . . . , xn, y] as the coordinate ring of An × A1, and IAn(X) = 〈f1(x), . . . , fr(x)〉, then X =Z(f1(y · x), . . . , fr(y · x)). There are two projections

X

An A1

πnπ1

.

The variety X is reducible, with two components X = X1 ∪ X2.

X1 = π−1n (A1 \ {0})

X2 = {(a, 0) : a ∈ An}.

Then we haveπn(X1) =

⋃q∈`

q, ` is secant to X, with 0 ∈ `.

that is, it is the closure of the set of all points lying on secants to X through our fixed point. The tangentcone to X at p = 0 is

Tp := πn(X1 ∩ π−11 (0)).

For a polynomial f ∈ K[x] we write f (m) for the homogeneous part of f of degreem. So f = f (m)+f (m+1)+· · ·+f (d). We call the homogeneous component of minimal degree the local leading form of f , and we denote it bylead(f). Given that 0 ∈ X, we have that the deg lead(f) > 0 for f ∈ I(X). So f(y · x) = ykf (k) + · · ·+ ydf (d).So we have

Tp = Z(lead(f) : f ∈ I(X)).

This is a cone because it is defined by homogeneous polynomials. We also have Tp ⊆ Θp, with equality if andonly if the point is smooth.For plane curves X = Z(f) ⊆ A2, and given that K = K, the form lead(f) splits as products of linear forms, sothe tangent cone splits as a union of lines. If the point is singular, then deg(f) > 1 is called the multiplicityof the singular point.

45

5.6 Local Parameters

We will study now what is so good about smooth points. So fix p ∈ X a nonsingular point of an affine varietywith dimpX = n.

Definition 5.11. Functions u1, . . . , un ∈ Op are local parameters at p if each ui ∈ mp and their equivalenceclasses form a basis of mp

/m2

p.

Being a system of parameters is equivalent to the linear forms dp u1, . . . ,dp un on Θp are linearly independent.This follows from the isomorphism dp : mp

/m2

p−→ Θ∗p. Since dimK Θp = n, this also means that the

homogeneous linear system of equations dp u1, . . . ,dp un only has the trivial solution.We may assume that X is affine and the ui are regular (otherwise, just replace the original with an affineneighborhood). Then each ui defines a hypersurface on X, say Yi := ZX(Ui) ⊆ X.We have

ΘYi,p = Li := ZΘp(dp ui) ⊆ ΘX,p.

We have by definition ΘYi,p ⊆ Li, and the other inclusion follows for dimension reasons and the fact that thelinear forms are not trivial. This implies that p is a nonsingular point of Yi as well. This implies that in someneighborhood of p the intersection of the Yi is exactly p. If there was a component of the intersection whichhad larger dimension, then the tangent space at to it at p would be contained in all the ΘYi,p, contradictingthe linear system dp ui having no nontrivial solutions.

Theorem 5.12. If u1, . . . , un are local parameters at p and assuming they are regular on X, with Yi = ZX(ui).Then p is a nonsingular point on each Yi and ⋂

ΘYi,p = 0.

Proof. See above.

Definition 5.13. Subvarieties Y1, . . . , Yr of a nonsingular variety X meet transversally at a point p ∈⋂Yi if

codimΘX,p

(⋂ΘYi,x

)=∑

codimX Yi.

For example, two curves on a nonsingular surface are transversal at a point of intersection if they are bothnonsingular there and their tangent lines are different.

Remark 5.14. If Yi meet transversally at p then⋂Yi is also nonsingular at p. Use the Definition 5.13 and

equation (4.1).

Theorem ?? shows that the sections by linear systems of parameters meet transversally.

Theorem 5.15. Local parameters generate the maximal ideal mp ⊆ Op.

Proof. This is the local version of Nakayama’s Lemma.

Definition 5.16. A local ring (O,m) is a regular ring if dimKrullO = dimO/m(m/m2).

In general, by the Nakayama Lemma, there only is a inequality “≤”. Nakayama + the description of the Krulldimension of a local ring as

dimKrullO = min{r : ∃u1, . . . , ur ∈ m such that mk ⊆ 〈u1, . . . , ur〉 for some k}.

(See Theorems 10.1 and 10.2 in the Algebra 1 Lecture Notes)

So p is nonsingular if and only if the local ring Op is nonsingular.

46

5.7 Completions

For any function f ∈ Op, set f(p) = α0, and f1 := f − α0. Then f ∈ mp. Let u1, . . . , un be a local system ofparameters at p. By definition, u1, . . . , un generate mp/m

2p. Thus, there exist α1, . . . , αn ∈ K such that

f2 := f1 −∑ni=1 αiui ∈ m2

p.

We can proceed by writing f2 =∑gihi, with gi, hi ∈ mp, and to each we can apply the same procedure:

subtract some linear combination of the s.o.p to obtain elements of m2p. Combining these get an

f3 = f − α0 −∑

αiui −∑

αjkujuk ∈ m3p.

So we can do this up to any k:

f −k∑i=0

Fi(u1, . . . , un) ∈ mk+1p .

with Fi ∈ K[x1, . . . , xn] forms of degree i. We would like this to go on forever, so we resort for this to the formalpower series ring K[[x1, . . . , xn]].

Definition 5.17. A formal power series Φ =∑i≥0 Fi ∈ K[[x1, . . . , xs]] is called the Taylor series of a function

f ∈ Op if for each k ≥ 0 we havef −

∑ki=0 Fi(u1, . . . , un) ∈ mk+1

p .

Example 5.18. The rational function 11−x at 0 ∈ A1 has the Taylor expansion

∑i≥0 x

i, beause

1

1− x−

k∑i=0

xi =xk+1

1− x∈ mk+1

0 .

The association depends on the local system of parameters however.

Theorem 5.19. Every function has at least one Taylor series (at p). If the point p ∈ X is also nonsingular,then the Taylor series is unique.

Proof. First part, by the above discussion.For the second, it is enough to show it for f = 0. Namely that if u1, . . . , un is a s.o.p at a nonsingular point,then Fk(u1, . . . , un ∈ mk+1

p implies F = 0 (as a form). See [Sha13, Theorem 2.7, p.102] for the details.

We have determined a map τ : Op −→ K[[x]], which takes f to its Taylor series. This is a homomorphism ofrings. If f is in the Kernel, then f ∈

⋂k≥0 m

kp = (0) (why? Nakayama!), so we get:

Theorem 5.20. A function f ∈ Op is uniquely determined by any of its Taylor series. In other words, Op is(isomorphic via τ to) a subring of K[[x]].

Theorem 5.21. If p is a nonsingular point of X, then there is a unique component of X containing it.

Proof. We may assume that X is already an affine neighborhood of p such that p belongs to all the irreduciblecomponents. Then we have

A(X) ⊆ OX,p ⊆ K[[x]]

The firs inclusion is by definition, the second by Theorem 5.20. As K[[x]] is a domain, we conclude.

Corollary 5.22. The singular locus of any (not nec. irreducible) algebraic variety is closed.

Proof. It is the union of the singular loci of the (finitely many) irreducible components with all the intersectionsof the irreducible components.

47

For a local ring Op we can compute the mp-adic completion, Op. We have

Op ⊆ Op,

and if p is nonsingular, then Op is just the formal power series ring. Otherwise, this ring is an importantcharacteristic of a singularity.If the completions of the local rings at p ∈ X and at q ∈ Y are isomorphic, then we say that the varieties Xand Y are formally analytically equivalent in neighbourhoods of these points.

5.8 Properties of Nonsingular points

Theorem 5.23. Any irreducible subvariety Y ⊆ X of codimension 1 has a local equation in a neighbourhoodof any nonsingular point p ∈ X.

Theorem 5.24. If X is nonsingular and ϕ : X −→ Pn is rational, then the set of points at which ϕ is notregular has codimension ≥ 2.

Corollary 5.25. If two nonsingular projective curves are birational, then they are isomorphic.

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Index

k-th exterior power, 21(relative) ideal of Y , 9(relative) zero set, 9

affine algebraic set, 6affine coordinate ring, 9affine rational variety, 13algebraic subsets of X, 9

bihomogeneous, 18birational, 13, 31birational equivalent, 13

codimension of Y in X, 32complete intersection, 33cotangent space, 43curve, 32

differential, 41–43dimension, 32domain of definition, 12, 31domain of regularity, 31

exterior algebra, 21

fibre, 37field of rational functions, 12finite map, 29formally analytically equivalent, 48function field, 16, 31function filed, 12

Grassmannian, 22

homogeneous coordinate ring, 14hyperplane, 18

ideal of Y , 7in general position, 19independent, 19intersection multiplicity, 40irreducible, 8irreducible components, 8isomorphism, 12

line, 18linear subspace, 18local leading form, 45local parameters, 46local property, 15local ring, 39local ring of X at a point P ∈ X, 13

multiplicity of the singular point, 45

Noetherian, 8nonsingular point, 44nonsingular variety, 44

Plücker coordinates, 22Plücker Embedding, 22Plücker relations, 25polynomial function, 9polynomial map, 11principal open set, 15product X × Y , 17projective algebraic set, 14proper morphism, 28pullback, 11pure tensor, 23

quasi-affine variety, 13quasi-projective variety, 15

rational map, 12, 31reducible, 8regular map, 25regular ring, 46

set theoretical complete intersection, 33singular point, 44smooth point, 44span of two linear subspaces, 18subvariety, 15surface, 32

tangent, 40tangent bundle, 44tangent cone, 45tangent space, 40, 43

49

Taylor series, 47transversally, 46

Zariski Topology, 6zero set, 6

50