Enumerativity of virtual Tevelev degrees

Lian, Carl; Pandharipande, Rahul

doi:10.48550/arxiv.2110.05520

Cited by 4 publications

(9 citation statements)

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“…Our results concern exact calculations of virtual Tevelev degrees in two main cases: cominuscule flag varieties and low degree complete intersections in projective spaces. An asymptotic equality between virtual and enumerative Tevelev degrees for certain Fano varieties (including flag varieties and low degree hypersurfaces) is proved in [LP21], so many of our calculations are actual curve counts. 1.3.…”

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confidence: 98%

“…Tevelev degrees have been studied in several contexts [CL21, CPS21, FL21, LP21, Tev20] starting with X = P 1 . Almost always, virtual Tevelev degrees are much better behaved than enumerative Tevelev degrees [LP21] and general Gromov-Witten invariants. Our results concern exact calculations of virtual Tevelev degrees in two main cases: cominuscule flag varieties and low degree complete intersections in projective spaces.…”

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confidence: 99%

“…We index here the curve classes of Q r ⊂ P r+1 for r ≥ 3 by their associated degree d in P r+1 . For fixed genus g, using the enumerativity results of [LP21], the virtual Tevelev degrees for Q r in sufficiently high degree d are actual curve counts.…”

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confidence: 99%

“…The precise statement in enumerative geometry from [LP21] is the following. Let C be a fixed general curve of genus g. The dimension constraint for Q r is d = n + g − 1 .…”

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confidence: 99%

“…when the dimension constraint is satisfied. For fixed genus g, using the enumerativity results of [LP21], the virtual Tevelev degrees for all flag varieties in sufficiently 5 See also [ABPZ21,Hu15] for recent results on the Gromov-Witten theory of complete intersections which also confront the primitive cohomology. 6 The full definition is reviewed in Section 3.…”

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Tevelev degrees in Gromov-Witten theory

Buch¹,

Pandharipande²

2021

Preprint

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For a nonsingular projective variety X, the virtual Tevelev degree in Gromov-Witten theory is defined as the virtual degree of the morphism from Mg,n(X, d) to the product Mg,n ×X n . After proving a simple formula for the virtual Tevelev degree in the (small) quantum cohomology ring of X using the quantum Euler class, we provide several exact calculations for flag varieties and complete intersections. In the cominuscule case (including Grassmannians, Lagrangian Grassmannians, and maximal orthogonal Grassmannians), the virtual Tevelev degrees are calculated in terms of the eigenvalues of an associated self-adjoint linear endomorphism of the quantum cohomology ring. For complete intersections of low degree (compared to dimension), we prove a product formula. The calculation for complete intersections involves the primitive cohomology. Virtual Tevelev degrees are better behaved than arbitrary Gromov-Witten invariants, and, by recent results of [LP21], are much more likely to be enumerative.

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confidence: 98%

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confidence: 99%

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confidence: 99%

“…The precise statement in enumerative geometry from [LP21] is the following. Let C be a fixed general curve of genus g. The dimension constraint for Q r is d = n + g − 1 .…”

mentioning

confidence: 99%

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confidence: 99%

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Tevelev degrees in Gromov-Witten theory

Buch¹,

Pandharipande²

2021

Preprint

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Linear Series on General Curves With Prescribed Incidence Conditions

Farkas¹,

Lian²

2022

J. Inst. Math. Jussieu

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Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree d and dimension r on a general curve of genus g satisfying prescribed incidence conditions at n points. We determine these numbers completely for linear series of arbitrary dimension when d is sufficiently large, and for all d when either $r=1$ or $n=r+2$ . Our formulas generalise and give new proofs of recent results of Tevelev and of Cela, Pandharipande and Schmitt.

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Gromov–Witten theory of complete intersections via nodal invariants

Argüz

Bousseau

Pandharipande

et al. 2023

Journal of Topology

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We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov-Witten theory are stated in complete generality and are of independent interest.

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Enumerativity of virtual Tevelev degrees

Cited by 4 publications

References 8 publications

Tevelev degrees in Gromov-Witten theory

Tevelev degrees in Gromov-Witten theory

Linear Series on General Curves With Prescribed Incidence Conditions

Gromov–Witten theory of complete intersections via nodal invariants

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