Tangent cones to generalised theta divisors and generic injectivity of the theta map

Hitching, George H.; Hoff, Michael

doi:10.1112/s0010437x17007539

Cited by 5 publications

(7 citation statements)

References 23 publications

(39 reference statements)

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“…1,g−1 (V) for stable vector bundles of rank r and vanishing determinant. Therefore, they also construct vector bundles with properties ( 1)-( 3) (see [15,Prop. 2.8] and Proposition 6.5 as its generalisation).…”

Section: Generatedness Of Petri General Bundlesmentioning

confidence: 99%

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Nonemptiness and smoothness of twisted Brill–Noether loci

Hitching

Hoff

Newstead

2020

Annali di Matematica

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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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Section: Generatedness Of Petri General Bundlesmentioning

confidence: 99%

“…which are furthermore globally generated (similar as in Sect. 6) was used in [15] to prove the generic injectivity of the theta map.…”

Section: Proposition 83mentioning

confidence: 99%

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Nonemptiness and smoothness of twisted Brill–Noether loci

Hitching

Hoff

Newstead

2020

Annali di Matematica

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“…Our interest is primarily in the infinitesimal geometry of

B_{r, d}^{k}

at singular points. As motivation, we note that in [14, 21], the infinitesimal geometry of generalised theta divisors associated to higher rank vector bundles (examples of twisted Brill–Noether loci ) was used to prove ‘Torelli‐type’ theorems (recovering the curve and the bundle, respectively). It seems therefore natural to investigate what can be recovered from the tangent cones

{scriptT}_{E} B_{r, d}^{k}

at singular points

E

.…”

Section: Introductionmentioning

confidence: 99%

A Riemann–Kempf singularity theorem for higher rank Brill–Noether loci

Hitching

2020

Bull. London Math. Soc.

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Given a vector bundle V of rank r over a curve X, we define and study a surjective rational map Hilb d (PV) Quot 0,d (V *) generalising the natural map Sym d X → Quot 0,d (OX). We then give a generalisation of the geometric Riemann-Roch theorem to vector bundles of higher rank over X. We use this to give a geometric description of the tangent cone to the higher rank Brill-Noether locus B k r,d at a suitable bundle E. This gives a generalisation of the Riemann-Kempf singularity theorem. As a corollary, if k = r and h 0 (X, E) = r + n, we show that the nth secant variety of the rank one locus of PEnd E is contained in the tangent cone to B r r,d .

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“…Our interest is primarily in the infinitesimal geometry of B k r,d at singular points. As motivation, we note that in [Pau03] and [HH17] the infinitesimal geometry of generalised theta divisors associated to higher rank vector bundles (examples of twisted Brill-Noether loci ) was used to prove "Torelli-type" theorems (recovering the curve and the bundle respectively). It seems therefore natural to investigate what can be recovered from the tangent cones T E B k r,d at singular points E. Note that [CT11] gives a comprehensive introduction and many interesting results on the singular loci of higher rank Brill-Noether loci and twisted Brill-Noether loci.…”

Section: Introductionmentioning

confidence: 99%

A Riemann--Kempf singularity theorem for higher rank Brill--Noether loci

Hitching

2017

Preprint

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Given a vector bundle V over a curve X, we define and study a surjective rational map Hilb d (PV ) Quot 0,d (V * ) generalising the natural map Sym d X → Quot 0,d (OX). We then give a generalisation of the geometric Riemann-Roch theorem to vector bundles of higher rank over X. We use this to give a geometric description of the tangent cone to the Brill-Noether locus B r r,d at a suitable bundle E with h 0 (E) = r + k. This gives a generalisation of the Riemann-Kempf singularity theorem. As a corollary, we show that the kth secant variety of the rank one locus of PEnd E is contained in the tangent cone.

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Tangent cones to generalised theta divisors and generic injectivity of the theta map

Cited by 5 publications

References 23 publications

Nonemptiness and smoothness of twisted Brill–Noether loci

Nonemptiness and smoothness of twisted Brill–Noether loci

A Riemann–Kempf singularity theorem for higher rank Brill–Noether loci

A Riemann--Kempf singularity theorem for higher rank Brill--Noether loci

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