Singularities of Tangent Varieties to Curves and Surfaces · Ian Porteous, Geometric...

Singularities of Tangent Varietiesto Curves and Surfaces

Goo Ishikawa

Department of Mathematics, Hokkaido University, Japan

Workshop on Singularities in Geometry and Applications

15th May to 21th May, 2011,

Stefan Banach International Mathematical Center

Bedlewo, Poland

Goo Ishikawa, Hokkaido University, Japan Singularities of Tangent Varieties to Curves and Surfaces 1

【 Tangent Variety】Embedded tangent spaces to a submanifold draw a variety inthe ambient space, which is called the tangent variety to thesubmanifold.

Example.Space curve Tangent variety


Example. Space curve γ : R → R3, γ(t) = (t, t2, t3).Parametrisation of tangent variety F : R2 → R3,F(t, s) := γ(t) + sγ′(t) = (t + s, t2 + 2st, t3 + 3st2)

Jacobi matrix of F : J(F) =

1 12t + 2s 2t

3t2 + 6st 3t2

rank J(F) < 3 ⇐⇒ s = 0. Take the transversal x1 = 0, then,s = −t, and we have x2 = −t2, x3 = −2t3, the planar cusp.


Tangent varieties appear in various geometric problems andapplications naturally.

— Developable (flat) surfaces (with zero Gaussian curvature).— Varieties with degenerate Gauss mapping.

— Varieties with degenerate projective dual.

— Bifurcation locus of projections.

— Relations to invariant theory.

— Geometric theory of differential equations.

— Examples of non-isolated singularities.————————————————————————Ian Porteous, Geometric Differentiation, for the intelligence of curves and

surfaces, Camb. Univ. Press. 1994.


Example. Let V4 = a0x3 + a1x2y + a2xy2 + a3y3 ∼= R4 be thespace of homogeneous polynomials of degree 3 with variablesx,y. The polynomials with triple zeros form a cubic curve Cin P(V) ∼= RP3. The tangent variety Tan(C) to C, called theumbilic bracelet, coincides with the set of polynomials withmultiple zeros.

The umbilic bracelet has cuspidal edge singularities along C:

Tan(C) is obtained from “deltoidal prism” by attaching by

120-rotation.


【 Local classification of tangent varieties to space curves】It is known that the tangent variety (tangent developable) to ageneric space curve has singularities each of which is locallydiffeomorphic to the cuspidal edge orto the folded umbrella (cuspidal cross cap) (Cayley, Cleave).

The cuspidal edge and the folded umbrella (cuspidal cross cap).

(The folded umbrella appears at an isolated point of zero “torsion”. )


If we consider a curve together with its osculating framings,we are led to the classification of tangent varieties to genericosculating framed curves (possibly with singularities in them-selves) in the three dimensional projective space.Then the list consists of 4 singularities: the cuspidal edge, thefolded umbrella and moreover the swallowtail and the Mondsurface (‘cuspidal beak to beak’).

swallowtail and Mond surface (cuspidal beak to beak)


————————————————————————In this talk we present the classification results ongeneric singularities of tangent varieties to curves(resp. osculating framed curves)with arbitrary codimension in projective spaces.————————————————————————————————————————————————The point of my talk isto relate the study of tangent varietiesto certain kinds of differential systems on flag manifolds.(The method initiated by Arnol’d and Scherbak).

————————————————————————


【 Differential Systems on Flag Manifolds】Let V be an (N + 2)-dimensional real vector space.Consider the complete flag manifold:

F (V) := V1 ⊂ V2 ⊂ · · ·VN+1 ⊂ V | dim(Vi) = i.(dimF = (N+1)(N+2)

2

).

We denote by πi : F → Gr(i,V) the canonical projection

πi(V1,V2, . . . ,VN+1) = Vi.

Define a subbundle C ⊂ TF , Cartan’s canonical system, by

v ∈ C(V1,...,VN+1) ⇐⇒ πi∗(v) ∈ TGr(i,Vi+1)(⊂ TGr(i,V)),

(1 ≤ i ≤ N).


We see that rank(C) = N + 1 and C is bracket generating.

A C∞ curve c : I →F (from an open interval I) is calledan C-integral curve if c′(t) ∈ Cc(t) for any t ∈ I.

A C-integral curve can be phrased as a C∞-family

c(t) = (V1(t),V2(t), . . . ,VN+1(t)) in F withddt

Vi(t) ⊂ Vi+1(t).

Vi(t) moves along Vi+1(t) at every moment.


For a C∞ curve γ : I → P(V) = RPN+1, if we consider Frenet-Serret frame (osculating projective moving frame),

(γ(t) = e0(t), e1(t), . . . , eN+1(t)) : I →GL(RN+2) = GL(N + 2,R),

then, setting

Vi(t) := 〈e0(t), e1(t), . . . , ei−1(t)〉R , (1 ≤ i ≤ N + 1),

we have the C-integral lifting γ : I →F of γ, for the projectionπ1 : F → P(V).

Fγ ↓

Iγ−→ P(V)

In this case, γ is called osculating-framed.


Theorem(Recalled). Let N + 1 = 3. For a generic C-integralcurve c : I → F in C∞-topology, the tangent variety to theosculating-framed curve γ = π1 c : I → P(V4) = RP3 ateach point is locally diffeomorphic to the cuspidal edge, thefolded umbrella (cuspidal cross cap), the swallowtail or tothe Mond surface.

cuspidal edge, folded umbrella, swallowtail, Mond surface


Theorem. (Higher codimensional case.) Let N + 1 ≥ 4.For a generic C-integral curve c : I →F in C∞-topology, thetangent variety to the osculating framed curve γ = π1 c :I → P(VN+2) = RPN+1 at each point is locally diffeomor-phic to a cuspidal edge, an open folded umbrella (cuspidalnon-cross cap), an open swallowtail or to an open Mondsurface.

Remark. For a generic curve γ : I → P(VN+2), the tangentvariety to γ at each point is locally diffeomorphic to a cuspidaledge or to an open folded umbrella , if N + 1 ≥ 4.


cuspidal edge and open folded umbrella in R4.

open swallowtail and open Mond surface in R4.


Example.(Higher codimensional “umbilic bracelet”.)Let VN+2 = a0xN+1 + a1xNy + · · ·+ aN xyN + aN+1yN+1 ∼=RN+2 be the space of homogeneous polynomials of degreeN + 1. The polynomials with zeros of multiplicity N + 1 forma curve C in P(V) ∼= RPN+1. The tangent variety Tan(C) toC coincides with the set of polynomials with with zeros ofmultiplicity ≥ N. The surface Tan(C) has cuspidal edge sin-gularities along C.


【 Opening procedure】The tangent variety to a curve in RPN+1 projects locally to thetangent variety to a curve in RP3 (osculating 3-space). Thetangent variety in RPN+1 can be regarded as an “opening” ofthat in RP3.The open swallowtail appears in many context. (e.g. Singular

Lagrangian variety (Arnol’d), Singular solution to PDE (Givental’)).

The open folded umbrella appears as a “frontal-symplecticsingularity” (Janeczko-I, Quarterly J. Math., 2003).


【 (Flat) Projective Structure and the Type of a Curve】M : m-dimensional C∞ manifold, M =

∪α Uα, ϕα : Uα → ϕα(Uα)⊂ Rm

a chart, ϕβ ϕα : ϕα(Uα ∩ Uβ) → ϕβ(Uα ∩ Uβ) is fractional linear:

(x1, . . . , xm) 7→(

a0j +∑i ai

j xi

a00+∑i ai

0xi

)1≤j≤m

.

For a sequence of natural numbers 1 ≤ a1 < a2 < · · · < am,We call γ : I → M is of type (a1, a2, . . . , am) at t0 ∈ I if there is a pro-jective local coordinates (x1, x2, . . . , xm) centred at γ(t) such that γ isrepresented asx1(t) = ta1 + o(ta1 ), x2(t) = ta2 + o(ta2 ), . . . , xm(t) = tam + o(tam )

If (a1, a2, . . . , am) = (1,2, . . . , m), then t = t0 is called an ordinary point

of γ.

Differential equation and genericity imply a restrictionon types of curves.


Lemma.(Getting local normal forms).In RP3,tangent variety of a curve of type (1,2,3) is locally diffeomor-phic to the cuspidal edges in R3,tangent variety of a curve of type (1,2,4) is locally diffeomor-phic to the folded umbrella,tangent variety of a curve of type (2,3,4) is locally diffeomor-phic to the swallowtails,tangent variety of a curve of type (1,3,4) is locally diffeomor-phic to the Mond surface.


Lemma.(Higher codimensional case.)In RPN+1, N ≥ 3,

(i) the tangent variety of a curve of type (1,2,3, a4, . . . , aN+1)is locally diffeomorphic to (the cuspidal edges in R3)×RN−2,(ii) the tangent variety of a curve of type

(1,2,4,5, . . . , N + 1, N + 2) is locally diffeomorphic to(the open folded umbrella in R4)×RN−3,(iii) the tangent variety of a curve of type

(2,3,4,5, . . . , N + 1, N + 2) is locally diffeomorphic to(the open swallowtail in R4)×RN−3,(iv) the tangent variety of a curve of type

(1,3,4,5, . . . , N + 1, N + 2) is locally diffeomorphic to(the open Mond surface in R4)×RN−3.


【 Frontal mappings】The key observation for the classification of singularities oftangent varieties is that tangent variety Tan(γ) to a curve γ offinite type is frontal.

Definition. Let ` < m. A C∞-mapping f : L` → Mm is calledfrontal if(i) Reg( f ) = x ∈ L | f : (L, x) → (M, f (x)) is an immersionis dense in L and(ii) there exists (uniquely) a C∞ mapping

f : L → Gr(`, TM) =∪

x∈M Gr(`, Tx M) satisfying

f (x) = f∗(Tx L), for x ∈ Reg( f ).


Remark.— If f is an immersion, then f is frontal.

— A wave-front is frontal.

— For a parametrisation F : (I × R, I × 0) → P(V) of thetangent variety to a curve of finite type is frontal: the lifting Fis obtained by taking osculating planes to the curves.


Moreover we give the generic diffeomorphism classificationof singularities on tangent varieties to contact-integral curves(resp. osculating framed contact-integral curves) in a contactprojective space. ( Joint work with Machida, Takahashi. )

Let V be a symplectic vector space of dimension 2n + 2.Consider the isotropic flag manifold:

FLag = V1 ⊂V2 ⊂ · · · ⊂Vn ⊂Vn+1 ⊂V |Vn+1 is Lagrangian.

Note that FLag is a finite quotient of U(n + 1),dim(FLag) = (n + 1)2

and that FLag(V) is embedded into F (V) by

(V1,V2, . . . ,Vn,Vn+1) 7→ (V1,V2, . . . ,Vn,Vn+1,Vsn , . . . ,Vs

2 ,Vs1 ),

completed by taking symplectic orthogonals.


Define a differential system E ⊂ TFLag by

v ∈ E(V1,...,Vn+1) ⇐⇒ πi∗(v) ∈ T Gr(i,Vi+1)(⊂ T IGr(i,V)),

(1 ≤ i ≤ n).IGr : isotropic Grassmannian, πi : FLag → IGr(i,V).Then rank(E) = n + 1 and E is bracket generating.

If n = 1, then we have dimFLag = 4 and E is an Engel struc-ture on FLag.

An E -integral curve c : I →FLag is a C∞ family

(V1(t),V2(t), . . . ,Vn(t),Vn+1(t))

of isotropic flags in the symplectic vector space V such thatddt

Vi(t) ⊂ Vi+1(t).


Remark. The projective space P(V) ∼= RP2n+1 has the canon-ical contact structure D ⊂ T(P(V)) :For V1 ∈ P(V) and for v ∈ TV1 P(V), we define

v ∈ DV1 ⇐⇒ π1∗(v) ∈ T(P(Vs1 ))(⊂ T(P(V))).

If c : I →FLag is an E -integral curve, then γ = π1 c : I →P(V) is a D-integral curve.

Theorem. (Machida-Takahashi-I, 2011)Let n = 1. For a generic E -integral curve c : I →FLag

(in C∞-topology), the tangent varieties to the osculatingframed Legendre curve γ = π1 c : I → P(V) ∼= RP3 is lo-cally diffeomorphic to the cuspidal edge, to the Mond sur-face or to the generic folded pleat.


cuspidal edge, Mond surface and generic folded pleat in R3.

Remark. The type of the curve γ is (1,2,3), (1,3,4) or (2,3,5).The local diffeomorphism class of the tangent variety Tan(γ)is determined if type(γ) = (1,2,3) or (1,3,4), but it is NOTdetermined if type(γ) = (2,3,5) and there are exactly two dif-feomorphism classes, generic one and non-generic one.


Remark. We have obtained also the generic classification ofsingularities of tangent varieties to π2 c : I → LG(V)in the Lagrangian Grassmannian.

Remark. The local contactomorphism class of Tan(γ) is de-termined if type(γ) = (1,2,3), but it is NOT determined iftype(γ) 6= (1,2,3).

Remark. Let 2n + 2 = 6. For a generic E -integral curve c : I →FLag (in C∞-topology), the type of osculating framed contact-integral curve γ = π1 c : I → P(V) ∼= RP5 at each point of Iis given by one of

(1,2,3,4,5), (1,2,4,5,6), (1,3,4,6,7), (2,3,4,5,7).


Singularities of Tangent Varieties toSurfaces.


【 Classification problem of tangent varieties to surfaces】

Example. Let V =

A =

a11 a12 a13

a12 a22 a23

a13 a23 a33

∣∣∣∣∣∣∣ 3 × 3, symmetric

,

the vector space of quadratic form of variables x,y,z. dim(V) = 6.Let S = P(rank(A) = 1) ⊂ P(V) ∼= RP5 be the Veronese surface.Then we see that

Tan(S) = S ∪ P(rank(A) = 2, semi-indefinite) ( P(V).


For a generic surface S ∈ RP5, tangent varieties Tan(S) areperturbed into

Therefore tangent variety Tan(S) to a generic surface S ⊂ RP5

is NEVER frontal.


Now, let us consider another type of flag manifold:

F1,3,5(R6) = V1 ⊂ V3 ⊂ V5 ⊂ R6.

The differential system W ⊂ T(F1,3,5(R6)) is defined by

v ∈W(V1,V3,V5) ⇐⇒ πi∗(v) ∈ T(Gr(i,Vi+2))(⊂ T(Gr(i,R6)),

i = 1,3. dim(F1,3,5(R6)) = 13,rank(W) = 8.

Theorem. Let S be a smooth surface of RP5. Then Thetangent variety Tan(S) is frontal if and only if S is the pro-jection of a W-integral surface by π1 : F1,3,5(R6) → RP5.


A Legendre surface S ⊂ RP5 (for the canonical contact struc-ture) lifts to a W-integral surface. Therefore we have:

Corollary. The tangent variety Tan(S) to a Legendre sur-face in RP5 is frontal.

Example. A point p of a Legendre surface S in RP5 is called anordinary point if there exists a local projective-contact coordinatesx1, x2, x3, x4, x5 and a C∞ local coordinates (u,v) of S centred p suchthat locally S is given by

x1 = u,x2 = v,x3 = 1

2 au2 + buv + 12 cv2 + higher order terms,

x4 = 12 bu2 + cuv + 1

2 ev2 + higher order terms,x5 = −( 1

6 au3 + 12 bu2v + 1

2 cuv2 + 16 ev3) + higher order terms,


with D = dx5 − x1dx3 − x2dx4 + x3dx1 + x4dx2 = 0, and

rank(

a b cb c e

)= 2.

Then Tan(S) is locally diffeomorphic to (D4-singularity in R3)×R2 inR5.


【 Further problems. 】

Problem. Classify the singularities of tangent varieties to generic sur-faces in RP5. Relate singularities of tangent variety to height function(or hight family).(cf. Mochida, Romero-Fuster, Ruas, Inflection points and nonsingularembeddings of surfaces in R5, Rockey Mount. J. Math., 2003. )

Problem. Classify the singularities of tangent varieties of projectionsin RP5 of generic W-integral surfaces in F1,3,5(R6).Classify the singularities of tangent varieties of generic Legendre sur-faces in RP5.

Thank you !Dziekuje !

【 Contact-cone Legendre-null duality】Let (V4,Ω) be a real vector space of dimension 4 with a sym-plectic form Ω. We consider Lagrange flag manifold:

F Lag

1,2 :=

(`, L)

∣∣∣∣∣ ` ⊂ L ⊂ V, ` : line,L : Lagrange plane

P(V) = Gr(1,V) := ` ⊂ V | ` : line : real projective 3-space.

LG(V) := L ⊂ V | L : Lagrange plane : Lagrange Grassman-nian.

FLag1,2

π1 π2

P(V) LG(V)

【 Contact structure and cone (conformal) structure】P(V) has the natural contact structure:D ⊂ TP(V) defined by, for ` ∈ P(V),

D` := T`(π1π−12 π2π−1

1 (`)) = T`P(`skew) ⊂ T`P(V)

LG(V) has the indefinite conformal structure:The null cone C ⊂ TLG(V) is defined by, for L ∈ LG(V),

CL := Tangent ConeL(π2π−11 π1π−1

2 (L))

where π2π−11 π1π−1

2 (L) = L′ ∈ LG(V) | L′ ∩ L 6= 0 isthe Schubert variety.

【 Engel structure】

The Lagrange flag manifold F Lag

1,2 has an Engel structureE ⊂ TF Lag

1,2 defined by, for (`, L) ∈ F Lag

1,2 ,

E(`,L) := T(`,L)(π−12 π2π−1

1 (`)) = T(`,L)(π−11 π1π−1

2 (`))= Ker(π1)∗ ⊕ Ker(π2)∗

In other word:

v ∈ TF Lag

1,2 , v ∈ E(`,L) ⇐⇒ π1∗(v) ∈ TP(L) ⊂ TP(V).

The derived system E2 := E + [E ,E ] is equal to the pull-backπ∗

1D of the contact structure D.

Let I be an open interval.

A C∞ map f : I → (F ,E) is called an Engel integral curveif f∗(TI) ⊂ E (⊂ TF ).

A C∞ map g : I → (P(V),D) is called a Legendre curveif g∗(TI) ⊂ D (⊂ TP(V)).

A C∞ map h : I → (LG(V),C) is called a null curveif h∗(TI) ⊂ C (⊂ TLG(V)).

f : I → (F ,E) is an Engel integral curve =⇒π1 f : I → (P(V),D) is a Legendre curve.

π2 f : I → (LG(V),C) is a null curve.

【 Projective structure and the type of a curve】M : m-dimensional C∞ manifold, M =

∪α Uα, ϕα : Uα → ϕα(Uα)⊂ Rm

a chart, ϕβ ϕα : ϕα(Uα ∩ Uβ) → ϕβ(Uα ∩ Uβ) is fractional linear:

(x1, . . . , xm) 7→(

a0j +∑i ai

j xi

a00+∑i ai

0xi

)1≤j≤m

.

For a sequence of natural numbers 1 ≤ a1 < a2 < · · · < am,

A curve γ : I → M is of type a = (a1, a2, . . . , am) at t0 ∈ I if there is a

projective local chart (x1, x2, . . . , xm) centred at γ(t0) such that γ is

represented as

x1(t) = ta1 + o(ta1 ), x2(t) = ta2 + o(ta2 ), . . . , xm(t) = tam + o(tam )

P(V),LG(V) and F Lag

1,2 have projective structures suchthat π1-fibres and their π2-images and π2-fibres and theirπ1-images are projective lines.

【 Tangent surfaces to Legendre curves and null curves】For a space curve in projective 3-space of finite type, we define the

tangent surface (or the tangent developable) to the curve as the sur-

face ruled by the tangent lines to the curve.

Lemma :Let f : I → (F Lag

1,2 ,E) be an Engel integral curve.Suppose both π1 f and π2 f are curves of finite type.

Then π1(π−12 (π2( f (I)))) is the tangent surface to π1( f (I))

ruled by tangent lines to π1( f (I)),

while π2(π−11 (π1( f (I)))) is the tangent surface to π2( f (I))

ruled by tangent lines to π2( f (I)).

After defining Engel integral jet spaces, we set:Σπ1,a := jr f (t0) ∈ Jr

E (I,F Lag

1,2 ) | π1 f : I → P(V) is of type aΣπ2,b := jr f (t0)∈ Jr

E (I,F Lag

1,2 ) | π2 f : I →LG(V) is of type b

Lemma :[Codimension formula]Σπ1,a 6= φ ⇐⇒ a3 = a1 + a2, codim(Σπ1,a) = a2 − 2.Σπ2,b 6= φ ⇐⇒ b3 = 2b2 − b1, codim(Σπ1,a) = b2 − 2.

Lemma :[Duality formula]Suppose, for an Engel integral curve f : I →F Lag

1,2 ,that type(π1 f ) = a and that type(π2 f ) = b. Then(b1,b2,b3) = (a2 − a1, a2, a3), (a1, a2, a3) = (b2 − b1,b2,b3).

Theorem.For a generic Engel integral curve f : I → F Lag

1,2 , in C∞-topology, the pair of types and the pair of singularities oftangent surfaces of π1 f ,π2 f are given by one of follow-ing:

(I) (1,2,3), (1,2,3), cuspidal-edge, cuspidal edge.

(II) (1,3,4), (2,3,4), Mond surface, swallowtail.

(III) (2,3,5), (1,3,5), generic folded pleat, Scherbak surface.

the Mond surface the swallowtail

the generic folded pleat the Scherbak surface

A cuspidal edge (resp. Mond surface, swallowtail, genericfolded pleat, Scherbak surface)is locally diffeomorphic to the germ of parametrized surface(R2,0) → (R3,0) exactly given by

cuspidal edge : (x, t) 7→ (x, − 12 t2 + xt, 1

3 t3 − 12 xt2),

Mond surface : (x, t) 7→ (x, − 13 t3 + 1

2 xt2, 14 t4 − 1

3 xt3),

swallowtail : (x, t) 7→ (x, 16 t3 − xt, − 1

4 t4 + xt2),

generic folded pleat : (x, t) 7→ (x, − 16 t3 + xt − 1

8 t4 + 12 xt2,

120 t5 − 1

6 xt3 + 124 t6 − 1

8 xt4),

Scherbak surface : (x, t) 7→ (x, 13 t3 − 1

2 xt2, − 15 t5 + 1

4 xt4).

Theorem.For a generic Engel integral curve f : I → F Lag

1,2 , in C∞-topology,the pair of types and the pair of singularities of tangent surfaces ofπ1 f , π2 f are given by one of following:

(I) (1,2,3), (1,2,3), cuspidal-edge, cuspidal edge.

(II) (1,3,4), (2,3,4), Mond surface, swallowtail.

(III) (2,3,5), (1,3,5), generic folded pleat, Scherbak surface.

Singularities of Tangent Varieties to Curves and Surfaces · Ian Porteous, Geometric...

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