Counting maximal subbundles via Gromov-Witten invariants

Let V be a vector bundle over a smooth curve C. In this paper, we study twisted Brill–Noether loci parametrising stable bundles E of rank n and degree e with the property that $$h^0 (C, V \otimes E) \ge k$$ h 0 ( C , V ⊗ E ) ≥ k . We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill–Noether loci are nonempty and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and “nontwisted” Brill–Noether loci. We describe the tangent cones to the twisted Brill–Noether loci. We end with an example of a general bundle over a general curve having positive-dimensional twisted Brill–Noether loci with negative expected dimension.

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“…been studied for a long time, dating back to [19]; for more recent work and all r, see [27]. For n > 1 , see [16,20,32, Theorem 0.3] and [7].…”

mentioning

confidence: 99%

Nonemptiness and smoothness of twisted Brill–Noether loci

Hitching

Hoff

Newstead

2020

Annali di Matematica

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“…Proof. The assertion follows from [12], § 4, p. 132, Theorem 4.2, where "k" (respectively, "r") corresponds to our r (respectively, n).…”

Section: Computation Via the Vafa-intriligator Formulamentioning

confidence: 90%

“…In the following, we review some facts concerning these invariants (cf. [11]; [23]; [12]). Denote by s N n,d C the moduli space of stable bundles on C of rank n and degree d (cf.…”

Section: Computation Via the Vafa-intriligator Formulamentioning

confidence: 99%

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An Explicit Formula for the Generic Number of Dormant Indigenous Bundles

Wakabayashi

2014

Publ. Res. Inst. Math. Sci.

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Abstract. A dormant indigenous bundle is an integrable P 1 -bundle on a proper hyperbolic curve of positive characteristic satisfying certain conditions. Dormant indigenous bundles were introduced and studied in the p-adic Teichmüller theory developed by S. Mochizuki. Kirti Joshi proposed a conjecture concerning an explicit formula for the degree over the moduli stack of curves of the moduli stack classifying dormant indigenous bundles. In this paper, we give a proof for this conjecture of Joshi.

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“…Degeneration arguments reduce many questions about the moduli of vector bundles on smooth curves to questions on nodal curves. This work will be useful in that direction too, and is in fact being used for this purpose [8].…”

Section: Introductionmentioning

confidence: 97%

Maximal Subsheaves of Torsion-Free Sheaves on Nodal Curves

Bhosle

2006

J. London Math. Soc.

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Let Y be a reduced irreducible projective curve of arithmetic genus g 2 with at most ordinary nodes as singularities. For a subsheaf F of rank r , degree d of a torsion-free sheaf E of rank r,where the minimum is taken over all subsheaves of E of rank r . For a fixed r , s r defines a stratification of the moduli space U (r, d) of stable torsion-free sheaves of rank r, degree d by locally closed subsets U r ,s . We study the nonemptiness and dimensions of the strata. We show that the general element in each nonempty stratum is a vector bundle and it has only finitely many (respectively unique) maximal subsheaves of rank r for s r (r − r )(g − 1) (respectively s < r (r − r )(g − 1)). We prove that the tensor product of two general stable vector bundles on an irreducible nodal curve Y is nonspecial.

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Counting maximal subbundles via Gromov-Witten invariants

Cited by 19 publications

References 16 publications

Nonemptiness and smoothness of twisted Brill–Noether loci

Nonemptiness and smoothness of twisted Brill–Noether loci

An Explicit Formula for the Generic Number of Dormant Indigenous Bundles

Maximal Subsheaves of Torsion-Free Sheaves on Nodal Curves

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