Non-standard analysisNotion of tangency

Getting a group

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Zariski Geometries, Lecture 2

Masanori Itai

Dept of Math Sci, Tokai University, Japan

August 30, 2011 at Kobe university

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Table of Contents

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1 Non-standard analysis

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2 Notion of tangency

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3 Getting a group

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

1. Non-standard analysis

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Elementary extensions of Zariski structures

Recall that if M is a Zariski structure and M ′ is an elementaryextenstion of M ′, we can make M ′ to be a Zariski structure aswell.

This makes it possible to introduce the notion of infinitesimalneiborhood of a point.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Specializations

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Definition (Def 2.2.2 )

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M � M∗, M ⊂ A ⊆ M∗

A mapπ : A → M is called a(partial) specialization if

for everya ∈ An and ann-ary M-closedSwith a ∈ S∗ = S(M∗)we haveπ(a) ∈ S.

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Remark

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Specialization is just a model-theoretic homomorphism.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Infinitesinal neighborhood

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Definition (Defn 2.2.18)

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M � M∗

π : M∗ → M , a universal specialization

For a ∈ M n,Va = π−1(a)

is called aninfinitesimal neighborhood of a.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

2. Notion of tangency

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Ampleness (AMP)

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Definition (AMP)

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Zariski structure C is ample if there is a two-dimensional,irreducible, faithful family L of curves on C2. L is locally isomorphicto an open subset of C2.

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Remark

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AMP ≡ non-locally modular≡ non-linear

We assume (AMP) for the rest of today’s talk.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Lemma 3.8.5

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Lemma (Lemma 3.8.5)

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There exists an irreducible, faithful, one-dimensional smooth familyN of curves through (a, b).

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Example 3.8.8. (p. 71)

Work with C ≺ C∗

I ={(u, v, x, y, z) :

ux + v(y− 1) + z(z− 1) = 0 andux2 + v(y− 1)z + z(z− 1) = 0

}

For each u, v ∈ C (u , 0∨ v , 0), put

gu,v = I (u, v,C3) = {(x, y, z) : (u, v, x, y, z) ∈ I }

gu,v passes through (x, y, z) = (0, 1, 0) and (x, y, z) = (0, 1, 1)

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Example 3.8.8. (p. 71) Cont’d

ux + v(y− 1) + z(z− 1) = 0 andux2 + v(y− 1)z + z(z− 1) = 0

Eliminating z, we haveuv2x(y− 1)2 + v3(y− 1)3 + u2x2(x− 1)2− uvx(x− 1)(y− 1) = 0

Projection of gu,v on the (x, y)-plane are curves through (0,1)with a nodal singularity at (0, 1)

gu,v is non-singular at both (0, 1 , 0) and (0, 1, 1)

gu,v defines two families of local functions:

g0u,v : V0 → V1,0 and g1

u,v : V0 → V1,1

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Example 3.8.8. (p. 71) Cont’d

ux + v(y− 1) + z(z− 1) = 0 andux2 + v(y− 1)z + z(z− 1) = 0

The first coordinates of the functions define the branches ofthe planar curves through (0,1) with the correspondingtrajectory z = 0 and z = 1.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Branches of a curve at a point

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Definition (Def 3.8.7)

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Let (a, b) ∈ C2. γ ⊂ V(a,b) is said to be abranch of a curve at(a, b) if

there are somec ∈ Cm−2, family of curvesG through(a, b) ^ c,and a curveg ∈ G such that

the cover(u, (x, y) ^ z) 7→ (u, x) is regular and unramifiedγ = {(x, y) ∈ Va,b) : ∃z ∈ Vc (g′, (x, y) ^ z) ∈ I }

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We define the notion of two curves g1, g2 are tangent at a point,written

g1T g2

such that

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Proposition (Prop 3. 8. 14)

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TFAE

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1 g1Tg2

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2 ∀x ∈ Va ∀g′1∈ Vg1 ∃g′

2∈ Vg2 (g′

1(x) = g′

2(x))

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3 ∀x ∈ Va ∀g′2∈ Vg2 ∃g′

1∈ Vg1 (g′

1(x) = g′

2(x))

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We define the notion of two curves g1, g2 are tangent at a point,written

g1T g2

such that

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Proposition (Prop 3. 8. 14)

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TFAE

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1 g1Tg2

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2 ∀x ∈ Va ∀g′1∈ Vg1 ∃g′

2∈ Vg2 (g′

1(x) = g′

2(x))

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3 ∀x ∈ Va ∀g′2∈ Vg2 ∃g′

1∈ Vg1 (g′

1(x) = g′

2(x))

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We define the notion of two curves g1, g2 are tangent at a point,written

g1T g2

such that

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Proposition (Prop 3. 8. 14)

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TFAE

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1 g1Tg2

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2 ∀x ∈ Va ∀g′1∈ Vg1 ∃g′

2∈ Vg2 (g′

1(x) = g′

2(x))

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3 ∀x ∈ Va ∀g′2∈ Vg2 ∃g′

1∈ Vg1 (g′

1(x) = g′

2(x))

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Intuitive idea

Two curves g1 and g2 are tangent at (a, b)

iff they have a common tangent line at (a, b)

iff they have the same derivative.

In Zariski structures we don’t have the notion of derivative. Hencewe use the notion of branches of a curve in order to say that twocurves have the same derivative.

Two curves are tangent at (a, b) iff they have the same branch at(a, b).

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Tangency is an equivalence relation

G(a,b) =⋃

Gc,I

where c is a trajectory and I incidence relation.

G(a,b) is a collections of infinitesimal pieces of smooth(regular) curves passing trough the point (a, b).

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Corollary (Cor. 3.8.18)

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T is an equivalence relation onG(a,b)

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

3. Getting a group

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Notion of tangency gives rise to a group

Consider the composition of local functionsVa → Va modulothe tangency

Compositon defines an associative operation, pre group ofjets.

Any Zariski pre-group can be extended to a group with apre-smooth structure on it. (Weil’s group chanks theorem)

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Notion of tangency gives rise to a group

Consider the composition of local functionsVa → Va modulothe tangency

Compositon defines an associative operation, pre group ofjets.

Any Zariski pre-group can be extended to a group with apre-smooth structure on it. (Weil’s group chanks theorem)

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Notion of tangency gives rise to a group

Consider the composition of local functionsVa → Va modulothe tangency

Compositon defines an associative operation, pre group ofjets.

Any Zariski pre-group can be extended to a group with apre-smooth structure on it. (Weil’s group chanks theorem)

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Composition of curves

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Definition (Def 4.1.3)

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g1 ∈ G1, g2 ∈ G2 with G1,G2 ⊂ G(a,b) define thecompositioncurveg−1

2◦ g1 as

{(x1, x2) ∈ C2 : ∃y, z1, z2 ((x1, y, z1) ∈ g1 ∧ (z2, y, x2) ∈ g−12

)}

Branch at(a, a)

(g−12◦ g1)(a,a) = {(x1, x2) ∈ Va × Va : x−1

2= g−1

2(g1(x1))}

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

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Lemma (Lemma 4.1.2)

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For every g1, g2 ∈ G(a,b);

g1 T g2 iff g−11

T g−12

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Lemma (Lemma 4.1.4)

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Tangency relation T is preserved by composition of branches ofsufficiently generic pairs of curves;

g1 T h1 ∧ g2 T h2 =⇒ g−12◦ g1 T h−1

2◦ h1

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Recall (AMP) and the irreducible, faithful, one-dimensional smoothfamily N of curves through (a, b) from Lemma 3.8.5.

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Lemma (Lemma 4.1.7)

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Given a generic pair (`1, `2, n1) ∈ N3, there is n2 ∈ N such that

n−11◦ `1 T n−1

2◦ `2

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Composition gives rise to a group operation

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Lemma (Lemma 4.1.8)

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Given a generic( f1, f2) ∈ Haa × Haa, there is a genericg ∈ Haa

such thatg is tangent to the compositionf1 ◦ f2.

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Remark

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This gives the group operation!

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Proof of Lemma 4.1.8 (1)

Fix an ` ∈ N generic over ( f1, f2). By Lemma 4.1.7, there aren1, n2 ∈ N such that

f1 T n−11◦ ` and f2T n−1

2◦ `.

Claim: f1 ◦ f2 T n−11◦ n2

Notice that (n1, n2) is a generic pair as well.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Proof of Lemma 4.1.8 (2)

Consider x ∈ Va and (n′1, n′

2) ∈ V(n1,n2). We need to find

( f ′1, f ′

2) ∈ V( f1, f2) such that

f ′1 ◦ f ′2(x) = n′−11 ◦ n′2(x).

For this we choose first f ′2∈ V f2 so that f ′

2(x) = y = `−1 ◦ ◦n′

2(x).

Next we choose f ′1∈ V f1 so that f ′

1(y) = n′−1

1◦ `(y).

Recall the criterion of tangency given by Prop 3.8.14.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

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Proposition (Prop 4.1.9 (pre-group of jets))

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There is a one-dimensional irreducible manifold U and aconstructible irreducible ternary relation P ⊆ U3 which is the graphof a partial map U2 → U and determines a partial Z-groupstructure on U.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

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Theorem (Thm 4.1.13 (Z-version of Weil’s group chanks thm))

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For any partial irreducible Z-group U, there is a connectedZ-group G and a Z-isomorphism between some dense openU′ ⊆ U and dense open G′ ⊆ G.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Getting a group from partial group U

Define G to be a semi-group of partial functions U → Ugenerated by shifts by elements in U.

For h, g ∈ G, h = g if h(u) = g(u) on an open subset of U.

The semi-group operation is defined by the composition offunctions.

With some arguments, we can identify G with the constructiblesort (U × U)/E where E is an equivalence relation;

(a1, a2) E (b1, b2) iff

dim{u ∈ U : ∃v,w, x ∈ U P(a2, u, v) ∧ P(a1, v, x)∧P(b2, u,w) ∧ P(b1,w, x)} = dim U

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Getting a group from partial group U

Define G to be a semi-group of partial functions U → Ugenerated by shifts by elements in U.

For h, g ∈ G, h = g if h(u) = g(u) on an open subset of U.

The semi-group operation is defined by the composition offunctions.

With some arguments, we can identify G with the constructiblesort (U × U)/E where E is an equivalence relation;

(a1, a2) E (b1, b2) iff

dim{u ∈ U : ∃v,w, x ∈ U P(a2, u, v) ∧ P(a1, v, x)∧P(b2, u,w) ∧ P(b1,w, x)} = dim U

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Getting a group from partial group U

Define G to be a semi-group of partial functions U → Ugenerated by shifts by elements in U.

For h, g ∈ G, h = g if h(u) = g(u) on an open subset of U.

The semi-group operation is defined by the composition offunctions.

With some arguments, we can identify G with the constructiblesort (U × U)/E where E is an equivalence relation;

(a1, a2) E (b1, b2) iff

dim{u ∈ U : ∃v,w, x ∈ U P(a2, u, v) ∧ P(a1, v, x)∧P(b2, u,w) ∧ P(b1,w, x)} = dim U

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Getting a group from partial group U

Define G to be a semi-group of partial functions U → Ugenerated by shifts by elements in U.

For h, g ∈ G, h = g if h(u) = g(u) on an open subset of U.

The semi-group operation is defined by the composition offunctions.

With some arguments, we can identify G with the constructiblesort (U × U)/E where E is an equivalence relation;

(a1, a2) E (b1, b2) iff

dim{u ∈ U : ∃v,w, x ∈ U P(a2, u, v) ∧ P(a1, v, x)∧P(b2, u,w) ∧ P(b1,w, x)} = dim U

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

Semi-group G is in fact a group.

This is a general fact about semi-groups without 0 being definablein stable structure.

If g ∈ G does not have an inverse, then

gn+1G ( g nG, for all n ∈ Z>0.

This defines the strict order property on the structure, contradictingstability.

We have now a group!

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We show that G is a Z-group. (1)

Let d = dim U2. Choosing generic pairwise independenta1, · · · , ad ∈ U we can show that

G = {a1, · · · , ad} · U−1 =

d⋃

i=1

{ai · v : v ∈ U}.

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We show that G is a Z-group. (2)

Each ai · U−1 can be identified with {ai} × U thus consideredas a manifold.

Partial bijections ai · U−1 → a j · U−1 defined by

(ai , v) 7→ (a j ,w) if ai · w = a j · v

The relation (ai , v) ↔ (a j ,w) defines a constructibleequivalence relation on

S =

d⋃

i=1

ai · U−1

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We show that G is a Z-group. (2)

Each ai · U−1 can be identified with {ai} × U thus consideredas a manifold.

Partial bijections ai · U−1 → a j · U−1 defined by

(ai , v) 7→ (a j ,w) if ai · w = a j · v

The relation (ai , v) ↔ (a j ,w) defines a constructibleequivalence relation on

S =

d⋃

i=1

ai · U−1

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We show that G is a Z-group. (2)

Each ai · U−1 can be identified with {ai} × U thus consideredas a manifold.

Partial bijections ai · U−1 → a j · U−1 defined by

(ai , v) 7→ (a j ,w) if ai · w = a j · v

The relation (ai , v) ↔ (a j ,w) defines a constructibleequivalence relation on

S =

d⋃

i=1

ai · U−1

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We show that G is a Z-group. (3)

The relation (ai , v) ↔ (a j ,w) defines a constructibleequivalence relation on

S =

d⋃

i=1

ai · U−1

The relation E and the ternary operation P corresponding themultiplications are closed.

View G as a manifold defined setwise as S/E. And G iscovered by smooth subsets ai ·U−1. The multiplication P on Gis closed.

Hence G is a Z-group!

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We show that G is a Z-group. (3)

The relation (ai , v) ↔ (a j ,w) defines a constructibleequivalence relation on

S =

d⋃

i=1

ai · U−1

The relation E and the ternary operation P corresponding themultiplications are closed.

View G as a manifold defined setwise as S/E. And G iscovered by smooth subsets ai ·U−1. The multiplication P on Gis closed.

Hence G is a Z-group!

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We show that G is a Z-group. (3)

The relation (ai , v) ↔ (a j ,w) defines a constructibleequivalence relation on

S =

d⋃

i=1

ai · U−1

The relation E and the ternary operation P corresponding themultiplications are closed.

View G as a manifold defined setwise as S/E. And G iscovered by smooth subsets ai ·U−1. The multiplication P on Gis closed.

Hence G is a Z-group!

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

We show that G is a Z-group. (3)

The relation (ai , v) ↔ (a j ,w) defines a constructibleequivalence relation on

S =

d⋃

i=1

ai · U−1

The relation E and the ternary operation P corresponding themultiplications are closed.

View G as a manifold defined setwise as S/E. And G iscovered by smooth subsets ai ·U−1. The multiplication P on Gis closed.

Hence G is a Z-group!

Masanori Itai Zariski Geometries, Lecture 2

Non-standard analysisNotion of tangency

Getting a group

.

Corollary (Cop 4.1.14)

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The groupJ of jets ata on the curveC generated byU = Haa/T is aconnected Z-group of dimension 1.

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Proposition (Prop 4.1.15, Reineck’s theorem)

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A one-dimensional connected group G is Abelian. In particular J isAbelian.

Tomorrow we start with this J i.e., an irreducibleZariski curve C with an Aberian group structure onit.

Masanori Itai Zariski Geometries, Lecture 2