A short proof of the Göttsche conjecture

Kool, Martijn; Shende, Vivek; Thomas, Robert Paul

doi:10.2140/gt.2011.15.397

Cited by 64 publications

(72 citation statements)

References 27 publications

(36 reference statements)

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“…This result without the irreducibility requirement is proved in [KST,Proposition 2.1] under the weaker assumption of δ-very ampleness.…”

Section: Counting Nodal Curvesmentioning

confidence: 78%

Reduced classes and curve counting on surfaces I: theory

Kool¹,

Thomas²

2014

Alg. Geom.

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130

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Abstract. We develop a theory of reduced Gromov-Witten and stable pair invariants of surfaces and their canonical bundles.We show that classical Severi degrees are special cases of these invariants. This proves a special case of the MNOP conjecture, and allows us to generalise the Göttsche conjecture to the non-ample case. In a sequel we prove this generalisation.We prove a remarkable property of the moduli space of stable pairs on a surface. It is the zero locus of a section of a bundle on a smooth compact ambient space, making calculation with the reduced virtual cycle possible. IntroductionMotivation. Fix a nonsingular projective surface S and a homology class β ∈ H 2 (S, Z). There are various ways of counting holomorphic curves in S in class β; in this paper we focus on Gromov-Witten invariants [Beh, LT] and stable pairs [PT1,Ott]. Since these are deformation invariant they must vanish in class β if there exists a deformation of S for which the Hodge type of β is not (1, 1). We can see the origin of this vanishing without deforming S as follows. For simplicity work in the simplest case of an embedded curve C ⊂ S with normal bundle N C = O C (C). As a Cartier divisor, C is the zero locus of a section s C of a line bundle L := O S (C), giving the exact sequenceThe resulting long exact sequence describes the relationship between first order deformations and obstructions H 0 (N C ), H 1 (N C ) of C ⊂ S, and the deformations and obstructionsThe resulting "semi-regularity map"S) takes obstructions to deforming C to the "cohomological part" of these obstructions. Roughly speaking, if we deform S, we get an associated obstruction in H 1 (N C ) to deforming C with it; its image in H 0,2 (S) is the (0, 2)-part of the cohomology class β ∈ H 2 (S) in the deformed complex structure. Thus it gives the obvious cohomological obstruction to deforming C: that β must remain of type (1, 1) in the deformed complex structure on S.In particular, when S is fixed, obstructions lie in the kernel ofMore generally, if we only consider deformations of S for which β remains (1, 1) then the same is true. And when h 0,2 (S) > 0 but H 2 (L) = 0, the existence of this trivial H 0,2 (S) piece of the obstruction sheaf guarantees that the virtual class vanishes.So it would be nice to restrict attention to surfaces and classes (S, β) inside the Noether-Lefschetz locus, 1 defining a new obstruction theory using only the kernel of the semi-regularity map.2 Checking that this kernel really defines an obstruction theory in the generality needed to define a virtual cycle -i.e. for deformations to all orders, over an arbitrary base, of possibly non-embedded 1 The locus of surfaces S for which β ∈ H 2 (S) has type (1, 1); for more details see [Voi, MP]. 2For embedded curves this means we use the obstruction space H 1 (L) to deforming sections of L. We have been able to remove the obstructions H 2 (O S ) to deforming L since the space of line bundles is smooth over the Noether-Lefschetz locus. Here we give quite a general construction using a mixture ...

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“…This result without the irreducibility requirement is proved in [KST,Proposition 2.1] under the weaker assumption of δ-very ampleness.…”

Section: Counting Nodal Curvesmentioning

confidence: 78%

Reduced classes and curve counting on surfaces I: theory

Kool¹,

Thomas²

2014

Alg. Geom.

Self Cite

130

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“…They were studied in detail by Dürr-Kabanov-Okonek [DKO07] in the context of Poincaré invariants (algebraic Seiberg-Witten invariants [CK13]). More recently, the stable pair invariants of surfaces have been employed in the context of curve counting problems [PT10, MPT10, KT14,KST11]. We define new invariants for the nested Hilbert scheme of curves and points on S. Our main application of these invariants is in the study of Donaldson-Thomas theory of local surfaces, that is carried out in [GSY17b].…”

Section: Introductionmentioning

confidence: 99%

Nested Hilbert schemes on surfaces: Virtual fundamental class

Gholampour

Sheshmani

Yau

2020

Advances in Mathematics

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We construct natural virtual fundamental classes for nested Hilbert schemes on a nonsingular projective surface S. This allows us to define new invariants of S that recover some of the known important cases such as Poincaré invariants of Dürr-Kabanov-Okonek and the stable pair invariants of Kool-Thomas. In certain cases, we can express these invariants in terms of integrals over the products of Hilbert scheme of points on S, and relate them to the vertex operator formulas found by Carlsson-Okounkov. The virtual fundamental classes of the nested Hilbert schemes play a crucial role in the Donaldson-Thomas theory of local-surface-threefolds that we study in [GSY17b].

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“…We may now proceed to prove the main theorem of this section, Theorem 3.7, concerning the shape of the node polynomials: Proof We assume r is such that

scriptL

is r ‐very ample. Hence, by Proposition 2.1 in , a general r ‐dimensional linear system

{double-struckP}^{r} \subset | scriptL |

contains a finite number of r ‐nodal curves, appearing with multiplicity 1, and all other curves are reduced with geometric genus strictly larger than

g - r,

where

2 g - 2 = L \cdot (L + {scriptK}_{S}) .

These curves are excluded from the counting by subtracting from

X_{1} \cdot ... \cdot X_{r}

the equivalence of the polydiagonals. Indeed, this operation takes care both of the excess contribution as well as the contribution from embedded, distinguished varieties.…”

Section: Shape Of Node Polynomialsmentioning

confidence: 99%

“…We assume r is such that L is r -very ample. Hence, by Proposition 2.1 in [12], a general r -dimensional linear system P r ⊂ |L | contains a finite number of r -nodal curves, appearing with multiplicity 1, and all other curves are reduced with geometric genus strictly larger than g − r, where 2g − 2 = L · (L + K S ). These curves are excluded from the counting by subtracting from X 1 · .…”

Section: Consider the Fiber Productmentioning

confidence: 99%

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Structure of node polynomials for curves on surfaces

Qviller

2014

Mathematische Nachrichten

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We provide a structural generalization of a theorem by Kleiman-Piene, concerning the enumerative geometry of nodal curves in a complete linear system |L | on a smooth projective surface S. Provided that r, the number of nodes, is sufficiently small compared to the ampleness of the linear system, we show that, under certain assumptions, the number of r -nodal curves passing through points in general position on S is given by a Bell polynomial in universally defined integers a i (S, L ), which we identify, using classical intersection theory, as linear, integral polynomials evaluated in four basic Chern numbers. Furthermore, we provide a decomposition of the a i as a sum of three terms with distinct geometric interpretations, and discuss the relationship between these polynomials and Kazarian's Thom polynomials for multisingularities of maps.

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A short proof of the Göttsche conjecture

Cited by 64 publications

References 27 publications

Reduced classes and curve counting on surfaces I: theory

Reduced classes and curve counting on surfaces I: theory

Nested Hilbert schemes on surfaces: Virtual fundamental class

Structure of node polynomials for curves on surfaces

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