Enumeration of singular hypersurfaces on arbitrary complex manifolds Ritwik Mukherjee

Abstract In this paper we obtain an explicit formula for the number of hypersurfaces in a compact complex manifold X (passing through the right number of generic points), that has a simple node, a cusp or a tacnode. The hypersurfaces belong to a linear system, which is obtained by considering a holomorphic line bundle L over X. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M , counted with a sign, is the Euler class of V evaluated on the fundamental class of M .

Contents 1 Introduction

1

2 Topological computations

2

3 Proof of transversality

4

4 Examples

7

1

Introduction

Enumeration of singular curves in P2 (complex projective space) is a classical problem in algebraic geometry. For certain singularities X, the authors in [7] and [1] use a purely topological method to obtain an explicit formula for the following question: Question 1.1. Let X be a codimension k-singularity.a How many degree d-curves are there in P2 , passing through (d(d + 3)/2 − k) generic points and having a singularity of type X? One of the powers of that topological method is that it generalizes to enumerating curves on any complex surface. Furthermore, the method generalizes to enumerating singular hypersurfaces on an arbitrary compact complex manifold of a given dimension. Before stating the main result of this paper, we give a couple of definitions: Definition 1.2. Let L −→ X be a holomorphic line bundle over an m-dimensional complex manifold X and f : X −→ L a holomorphic section. A point q ∈ f −1 (0) is of singularity type Ak if there exists a

By codimension we mean the number of conditions having that particular singularity imposes on the space of curves. For example a node is a codimension one singularity, a cusp is a codimesnion two singularity, a taconde is a codimension three singularity and so on.

1

a coordinate system (z1 , z2 , . . . , zm ) : (U , q) −→ (Cm , 0) such that f −1 (0) ∩ U is given by z1k+1 +

m X

zi2 = 0.

i=2

Definition 1.3. A holomorphic line bundle L −→ X over a compact complex manifold X is sufficiently k-ample if L ≈ L⊗n 1 −→ X, where L1 −→ X is a very ample line bundle and n ≥ k. The following theorem is the main result of this paper: Theorem 1.4. Let X be an m-dimensional compact complex manifold and L −→ X a holomorphic line bundle. Let D := PH 0 (X, L) ≈ PδL ,

c1 := c1 (L)

and

xi := ci (T ∗ X)

where ci denotes the ith Chern class. Denote N (X) to be the number of hypersurfaces in X, that belong to the linear system H 0 (X, L), passing through δL − k generic points and having a singularity of type X, where k is the codimension of the singularity X. Then N (A1 ) = N (A2 ) = N (A3 ) =

m X (m + 1 − i)xi cm−i , 1

i=0 m X

 m−1 X  m + 1 − i m+2−i m−i 2 x1 xi cm−i−1 , m xi c1 + 1 2 2

i=0 m−2 X



(2)

i=0

t2 cm−i−2 xi + 1

m−1 X

t1 cm−i−1 xi + 1

i=0

i=0

where

(1)

m X

t0 cm−i xi , 1

(3)

i=0

  m + 1 − i  c21 t2 := (3m2 − m) + c1 x1 (6m − 1) + 6x21 , 3 2    m+1−i  t1 := c1 (3m2 − m) + x1 (6m − 1) and 2   m + 1 − i  3m2 m  t0 := − 1 2 2 

provided the line bundle L −→ X is sufficiently 2-ample, 3-ample and 4-ample respectively. Remark 1.5. In equations (1) to (3) we are making an abuse of notation; our intended meaning is the rhs evaluated on the fundamental class [X].

2

Topological computations

The main tool we use is the following fact from differential topology (cf. [3], Proposition 12.8): Theorem 2.1. Let V −→ X be an oriented vector bundle over a compact manifold X and s : X −→ V a smooth section that is transverse to the zero set. Then the Poincar´e dual of [s−1 (0)] in X is the Euler class of V . In particular, if the rank of V is same as the dimension of X, then the signed cardinality of s−1 (0) is the Euler class of V , evaluated on the fundamental class of X, i.e., | ± s−1 (0)| = he(V ), [X]i. 2

We are now ready to give a proof of formulas (1), (2) and (3). Let q1 , q2 , . . . , qδL −1 be δL − 1 generic points in X. Define Hi := {[f ] ∈ D : f (qi ) = 0}

ˆ i := Hi × X. H

and

Then N (A1 ) is the cardinality of the set ˆ1 ∩ ... H ˆ δ −1 . {([f ], q) ∈ D × X : f (q) = 0, ∇f |q = 0} ∩ H L

(4)

Let us now define sections of the following bundles: ∗ ψA0 : D × X −→LA0 := γD ⊗ L,

ψA1 :

−1 (0) ψA 0

−→VA1 :=

∗ γD

{ψA0 ([f ], q)}(f ) := f (q),

∗

⊗ T X ⊗ L,

{ψA1 ([f ], q)}(f ) := ∇f |q .

(5) (6)

Here γD is the tautological line bundle over D. In (5), the rhs is an element of the vector space Lq , the fiber of the line bundle L over q. Hence ψA0 is a section of LA0 . Similarly, the rhs of (6) is an element of Tq∗ X ⊗ Lq . Hence ψA1 is a section of VA1 . In section 3 we show that the sections ψA0 and ψA1 are transverse to the zero set if L −→ X is sufficiently 1-ample. Since the δL − 1 points are generic, we conclude that the intersection in (4) is transverse. Furthermore, if ([f ], q) belongs to the set defined in (4), then f has a genuine A1 -node at q (as opposed to something more degenerate like an A2 -node for instance). A rigorous proof of this claim is available in [2] for the special case when X := P2 and L := γP∗d2 , (if d ≥ 2). The same proof goes through for any X if the line bundle L −→ X is sufficiently 2-ample; hence we omit the proof in this paper. However, a self contained proof of this claim can be found in [5] for any L −→ X. The basic idea is to use the Families Transversality Theorem and Bertini’s Theorem. Next, we note that ψA0 and ψA1 are holomorphic. Hence, −1 (0)]i = he(LA0 )e(VA1 )e(γD )δL −1 , [D × X]i. N (A1 ) = he(VA1 )e(γD )δL −1 , [ψA 0

(7)

−1 (0)] in D × X is e(LA0 ). The second equality follows from the fact that the Poincar´e Dual of [ψA 0 Using the splitting principal, (7) simplifies to (1).

Next, we will give a proof of (2). Let PT X denote the projectivization of T X and γˆ −→ PT X the tautological line bundle over PT X. Then N (A2 ) is the cardinality of the set −1 ˜1 ∩ ... H ˜ δ −2 , (0), ∇2 f |q (v, ·) = 0 ∀ v ∈ lq } ∩ H {([f ], lq ) ∈ D × PT X : (f, q) ∈ ψA L 1

(8)

˜ i := π −1 (H ˆ i ) and π : D × PT X −→ D × X is the projection map. We now define a section where H of the following bundle −1 ∗ (0) −→ VPA2 := γˆ ∗ ⊗ γD ⊗ T ∗ X ⊗ L, ΨPA2 : π ∗ ψA 1

{ΨPA2 ([f ], lq )}(v ⊗ f ) := ∇2 f |q (v, ·)

given by

∀ v ∈ lq .

(9)

Note that the rhs of (9) belongs to Tq∗ X ⊗ Lq . Hence ΨPA2 is a section of VPA2 . We show in section 3, that this section is transverse to the zero set. Since the δL − 2 points are generic, the intersection in (8) is transverse. Since ΨPA2 is holomorphic, we conclude that −1 ∗ δL −2 (0)]i N (A2 ) = he(VPA2 )e(γD ) , [π ∗ ψA 1 ∗ δL −2 = he(π ∗ LA0 )e(π ∗ VA1 )e(VPA2 )e(γD ) , [D × PT X]i.

3

(10)

We now use the ring structure of H ∗ (PT X, Z) ([3], pp. 270) to conclude that λm − x1 λm−1 + x2 λm−2 + . . . + (−1)m xm = 0,

(11)

where λ := c1 (ˆ γ ∗ ). Using the splitting principal, (10) combined with (11) simplifies to (2). Finally, we are ready to prove (3). First, we note that N (A3 ) is the cardinality of the set 3 ˜ ˜ {([f ], lq ) ∈ D × PT X : ([f ], lq ) ∈ Ψ−1 PA2 (0), ∇ f |q (v, v, v) = 0 ∀ v ∈ lq } ∩ H1 ∩ . . . HδL −3 .

Let us now define a section of the following bundle ∗ ΨPA3 : Ψ−1 ˆ ∗3 ⊗ γD ⊗ L, PA2 (0) −→ LPA3 := γ

{ΨPA3 ([f ], lq )}(v

⊗3

3

⊗ f ) := ∇ f |q (v, v, v)

given by ∀ v ∈ lq .

(12)

Note that the rhs of (12) is an element of Lq . Hence ΨPA3 is a section of LPA3 . In section 3 we show that this section is transverse to the zero set. Since the δL − 3 points are generic and ΨPA3 is holomorphic, we conclude that ∗ δL −3 N (A3 ) = he(LPA3 )e(γD ) , [Ψ−1 PA2 (0)]i ∗ δL −3 = he(π ∗ LA0 )e(π ∗ VA1 )e(VPA2 )e(LPA3 )e(γD ) , [D × PT X]i.

(13)

Using the splitting principal, equations (13) combined with (11) simplifies to (3).

3

Proof of transversality

We now prove that the relevant sections sections are transverse to the zero set, provided the line bundle is sufficiently ample. We start with the following fact about very ample line bundles: Proposition 3.1. Let X be a compact complex m-manifold and L −→ X be a very ample line bundle. Let q ∈ X and v1 , . . . vm be a basis for T Xq . Then there exists sections s1 , . . . sm ∈ H 0 (X, L) such that for all i, j ∈ {1, 2 . . . m} si (q) = 0,

∇si |q (vi ) 6= 0

and

∇si |q (vj ) = 0

if i 6= j.

(14)

Proof: Since L −→ X is very ample, X embeds inside Pn for some n and L is the restriction of the hyperplane bundle γP∗n −→ Pn . Let e1 , . . . en be a basis for T Pn |q . It is easy to see that there exists sections ω1 , . . . ωn ∈ H 0 (Pn , γP∗n ) such that for all i, j ∈ {1, 2 . . . n} ωi (q) = 0,

∇ωi |q (ei ) 6= 0

and

∇ωi |q (ej ) = 0

if i 6= j.

∗ ∈ T ∗ X ⊗ L be a collection of m vectors such that Let v1∗ , . . . , vm q q

vi∗ (vi ) 6= 0

and

vi∗ (vj ) = 0

if i 6= j.

An element of T ∗ Xq ⊗ Lq can be thought of as an element of T ∗ Pnq ⊗ Lq ; choose a splitting of T Pn |q ≈ T Xq ⊕ T Pnq /T Xq and extend the linear functional on T Xq to the whole of T Pnq by declaring it to be zero on the quotient space T Pnq /T Xq . Since ∇ω1 |q , . . . ∇ωn |q forms a basis for T ∗ Pnq ⊗ Lq , we conclude that for all i ∈ {1, . . . , m} vi∗ = αi1 ∇ω1 |q + . . . αin ∇ωn |q 4

for some complex numbers αij . Define si := αi1 ω1 + . . . + αin ωn ,

∀ i ∈ {1, . . . , m}.

It is easy to see that si defined above satisfy equation (14). Corollary 3.2. Let X be a compact complex m-manifold and L −→ X be a very ample line bundle. Given any point q ∈ X, there exists sections s1 , . . . , sm ∈ H 0 (X, L) such that s1 (q) = 0, . . . , sm (q) = 0 and ∇s1 |q , . . . , ∇sm |q are linearly independent in T ∗ Xq ⊗ Lq . Proof: Immediate from Proposition 3.1. We will also need the following fact from ordinary calculus: Proposition 3.3. Let f1 , . . . , fm : Cm −→ C and g : Cm −→ C be a collection of holomorphic functions defined on a neighborhood of some point q ∈ Cm , such that ∂fi ∂fi fi (q) = 0, 6= 0, = 0 if i 6= j ∀i, j ∈ {1, . . . , m} and g(q) 6= 0. ∂zi z=q ∂zj z=q

Define hi : Cm −→ C to be hi := g · f1 · fi and j : Cm −→ C to be j := g · f13 , where · denotes pointwise multiplication. Let v := α1 ∂z1 + . . . αm ∂zm ∈ T Cm |q for some complex numbers αi , such that α1 6= 0. Then ∇hi |q = 0, ∇j|q = 0,

∇2 hi |q (v, ∂zi ) 6= 0, 2

∇ j|q (v, ·) = 0

∇2 hi |q (v, ∂zj ) = 0

if i 6= j and j 6= 1,

3

∇ j|q (v, v, v) 6= 0.

and

∗ m In particular, the vectors {∇2 hi |q (v, ·)}m i=1 are linearly independent in T C |q .

Proof: Follows from a repeated application of the product rule of ordinary calculus. Lemma 3.4. The sections ψA0 , ψA1 , ΨPA2 and ΨPA3 defined in (5), (6), (9) and (12) are transverse to the zero set, if the line bundle L −→ X is sufficiently 0-ample, 1-ample, 2-ample and 3-ample respectively. Proof: First we will show that ψA0 is transverse to the zero set. It suffices to show that the induced section   ψ˜A0 : H 0 (X, L) − 0 × X −→ L, ψ˜A0 (f, q) := f (q) is transverse to the zero set. Suppose ψ˜A0 (f, q) = 0. Since L −→ X is sufficiently 0-ample, the linear system H 0 (X, L) is base there exists a section η ∈ H 0 (X, L) such that η(q) 6= 0.  point free. Hence,  Consider the curve in H 0 (X, L) − 0 × X given by γ(t) := (f + tη, q). Transversality follows by observing ∇ψ˜A0 |(f,q) (γ ′ (0)) = η(q) 6= 0. Next we show that ψA1 is transverse to the zero set. It suffices to show that the induced section −1 (0) −→ T ∗ X ⊗ L, ψ˜A1 : ψ˜A 0

5

ψ˜A1 (f, q) := ∇f |q

is transverse to the zero set. Suppose ψ˜A1 (f, q) = 0. Since the line bundle L −→ X is sufficiently 1-ample, it is of the form L = L1 ⊗ L2 , where L1 is very ample and H 0 (X, L2 ) is base point free. By Corollary 3.2 there exists sections s1 , . . . sm ∈ H 0 (X, L1 ) such that si (q) = 0 and ∇si |q are linearly independent in T ∗ Xq ⊗ L1 |q . Let ξ ∈ H 0 (X, L2 ) be a section, such that ξ(q) 6= 0. Define −1 (0) given by γi (t) := (f + tηi , q). ηi := si ⊗ ξ ∈ H 0 (X, L). Consider the m distinct curves in ψ˜A 0 Transversality now follows by observing that ∇ψ˜A1 |(f,q) (γi′ (0)) = ∇ηi |q and the fact that ∇ηi |q are linearly independent in T ∗ Xq ⊗ Lq . Next we show that ΨPA2 is transverse to the zero set. It suffices to show that the induced section ˜ PA : π ∗ ψ˜−1 (0) −→ γˆ ∗ ⊗ T ∗ X ⊗ L, Ψ 2 A1

˜ PA (f, lq )}(v) := ∇2 f |q (v, ·) ∀ v ∈ lq {Ψ 2

is transverse to the zero set, where π : H 0 (X, L) × PT X −→ H 0 (X, L) × X is the projection map. ˜ PA (f, lq ) = 0. Since the line bundle L −→ X is sufficiently 2-ample, it is of the form Suppose Ψ 2 ⊗2 L = L1 ⊗ L2 , where L1 is very ample and H 0 (X, L2 ) is base point free. Let v1 , . . . vm be a basis for T Xq and s1 , . . . sm be the corresponding sections in H 0 (X, L1 ) that we get from Proposition 3.1. Let ξ ∈ H 0 (X, L2 ) be a section such that ξ(q) 6= 0. Given a vector v ∈ lq , let us assume without loss of generality that it can be written as v = α1 v1 + α2 v2 + . . . αm vm ,

α1 6= 0

for some complex numbers αi . Define ηi := s1 ⊗ si ⊗ ξ ∈ H 0 (X, L). Consider the m distinct curves −1 in π ∗ ψ˜A (0) given by γi (t) := (f + tηi , lq ). Observe that 1 ˜ PA |(f,l ) (γi′ (0))}(v) = ∇2 ηi |q (v, ·). {∇Ψ 2 q By writing the sections ηi in a particular trivialization and using Proposition 3.3, we conclude that ∇2 ηi |q (v, vi ) 6= 0

and

∇2 ηi |q (v, vj ) = 0

if i 6= j and j 6= 1.

This means that ∇2 ηi |q (v, ·) are linearly independent in T ∗ Xq ⊗ Lq and hence transversality follows. Finally, we show that ΨPA3 is transverse to the zero set. It suffices to show that the induced section ˜ PA : Ψ ˜ −1 (0) −→ γˆ ∗3 ⊗ T ∗ X ⊗ L, Ψ 3 PA2

˜ PA (f, lq )}(v ⊗3 ) := ∇3 f |q (v, v, v) ∀ v ∈ lq {Ψ 3

˜ PA3 (f, lq ) = 0. Since the line bundle L −→ X is sufficiently is transverse to the zero set. Suppose Ψ ⊗3 3-ample, it is of the form L = L1 ⊗ L2 , where L1 is very ample and H 0 (X, L2 ) is base point free. Let v1 , . . . vm be a basis for T Xq and s1 , . . . sm be the corresponding sections in H 0 (X, L1 ) that we get from Proposition 3.1. Let ξ ∈ H 0 (X, L2 ) be a section such that ξ(q) 6= 0. Given a vector v ∈ lq , let us assume without loss of generality, that it can be written as v = α1 v1 + α2 v2 + . . . αm vm ,

α1 6= 0

0 ˜ −1 for some complex numbers αi . Define η := s⊗3 1 ⊗ ξ ∈ H (X, L). Consider the curve in ΨPA2 (0) given ˜ −1 (0), follows from Proposition by γ(t) := (f + tη, lq ). The fact that this is indeed a curve in Ψ PA2 3.3. Transversality now follows, since by Proposition 3.3 we conclude that

˜ PA |(f,l ) (γ ′ (0))}(v ⊗3 ) = ∇3 η|q (v, v, v) 6= 0. {∇Ψ 3 q 6

Remark 3.5. For the formula for N (A1 ) to hold, we not only require that the sections ψA0 and ψA1 be transverse to the zero set, but also the fact that the intersections in (4) are transverse. It is to ensure this second fact that we require L −→ X to be sufficiently 2-ample. Similarly, for the hyperplanes to intersect transversally in the computations of N (A2 ) and N (A3 ) we require L −→ X to be sufficiently 3-ample and 4-ample respectively. The crucial fact we use is as follows: we consider the space M (q) ⊂ D, the closure of the space of curves that have an Ak -node at a specific point q. We then argue that the base locus of the linear system M (q) is exactly q using Bertini’s Theorem. This is where we need the fact that the line bundle is sufficiently (k + 1)-ample. The details of the proof (and how it implies that the hyperplanes intersect transversaly) can be found in [5].

4

Examples

Example 4.1. Let X be a surface and L −→ X a sufficiently 2-ample line bundle. Then (1) simplifies to N (A1 ) = 3c21 + 2c1 x1 + x2 ,

(15)

which agrees with the result of Vainsencher in [6]. Example 4.2. Let X := Pm and L := γP∗dm . Then     m + 1 m−i−1 i m+1 m−i m−i−1 i+1 xi cm−i = (−1) d and x x c = (−1) (m + 1) d . 1 i 1 i i

(16)

Equations (1), (2) and (3) combined with (16) imply that N (A1 ) = (m + 1)(d − 1)m , m(m + 1)(m + 2) N (A2 ) = (d − 1)m−1 (d − 2), 2 m(m + 1)(m + 2) (d − 1)m−2 (m2 d2 + m1 d + m0 ), N (A3 ) = 12 where m2 := 3m2 + 8m − 3, m1 := −12m2 − 28m + 8 and

(17) (18) (19) m0 := 12m2 + 24m − 12,

provided d ≥ 2, 3 and 4 respectively. For m = 2 this gives us N (A1 ) = 3(d − 1)2 , N (A2 ) = 12(d − 1)(d − 2), N (A3 ) = 50d2 − 192d + 168,

(20)

which recovers the formulas obtained in [7] and [1]. For general m, the numbers N (A2 ) and N (A3 ) agree with the results of Kerner in [4].b Example 4.3. Let X := P1 × P1 and L := π1∗ γP∗d1 1 ⊗ π2∗ γP∗d1 2 . Then c21 = 2d1 d2 , c1 x1 = −2(d1 + d2 ), x21 = 8, x2 = 4.

(21)

Equation (21), combined with (1), (2) and (3) gives N (A1 ) = 6d1 d2 − 4(d1 + d2 ) + 4,

(22)

N (A2 ) = 24(d1 − 1)(d2 − 1)

(23)

and

N (A3 ) = 100d1 d2 − 128(d1 + d2 ) + 156, provided d1 and d2 are both greater than or equal to 2, 3 and 4 respectively. b

The number N (A1 ) is not explicitly stated in [4].

7

(24)

Remark 4.4. Strictly speaking, (22), (23) and (24) hold when d1 and d2 are greater than 2, 3 and 4 respectively. However, the formulas for N (A1 ), N (A2 ) and N (A3 ) are valid when d1 = 1 and d2 = 1. There are indeed two curves of type (1, 1) through two generic points that have a node; there are no curves of type (1, 1) with either a cusp or a tacnode. Similarly, the formulas for N (A1 ) and N (A2 ) in (20) are strictly speaking valid when d ≥ 2 and 3 repsectively. However, the formula for N (A1 ) is valid even when d = 1 since there are no lines with a node and the formula for N (A2 ) is valid even when d = 1 or 2 since there are no lines or conics with a cusp. Hence, the conditions we have given for transversality are sufficient; these examples show they aren’t always necessary. Acknowledgements. The author thanks Indranil Biswas for suggesting that the topological method be applied to enumerate singular hypersurfaces and for suggesting example 4.3. Furthermore, the author is also grateful to Somnath Basu, Shane D’Mello and Vamsi Pingali for relevant discussions and comments about this paper.

References [1] S. Basu and R. Mukherjee, Enumeration of curves with one singular point. available at http://arxiv.org/abs/1308.2902. [2]

, Enumeration of curves with singularities: Further https://www.sites.google.com/site/ritwik371/home.

details.

available

at

[3] R. Bott and L. W. Tu, Differential forms in algebraic topology, vol. 82 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982. [4] D. Kerner, Enumeration of uni-singular algebraic hypersurfaces, Proc of London Mathematical Society, 3 (2008), pp. 623–668. [5] R. Mukherjee, Enumeration of singular hypersurfaces: general position arguments. available at https://www.sites.google.com/site/ritwik371/home. [6] I. Vainsencher, Enumeration of n-fold tangent hyperplanes to a surface, J. Algebraic Geom., 4 (1995), pp. 503–526. [7] A. Zinger, Counting plane rational curves: http://arxiv.org/abs/math/0507105. DEPARTMENT

OF

old and new approaches.

MATHEMATICS, TIFR, TATA INSTITUTE OF

400005, INDIA

E-mail address : [email protected]

8

available at

FUNDAMENTALR ESEARCH,

MUMBAI