Singularities of Brill‐Noether loci for vector bundles on a curve

Casalaina-Martin, Sebastian; Bigas, Montserrat Teixidor i

doi:10.1002/mana.200910093

Cited by 16 publications

(19 citation statements)

References 28 publications

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“…By Theorem 2.2, for each such

normalΠ

there exists

Z_{normalΠ} \in {normalHilb}^{d} false(double-struckP E^{*} false)

such that

{scriptO}_{X} \otimes Π = {(E^{*})}_{Z_{Π}}

. Combining this observation with results from [6], we obtain the following generalisation of the Riemann–Kempf singularity theorem. In what follows, for any

ℓ ⩾ r

we denote by

U^{ℓ}

the open subset of

Gr (ℓ, H^{0} false(X, E false))

of subspaces which generically generate

E

.…”

Section: Introductionmentioning

confidence: 51%

“…In [6, § 5], it is shown that the tangent cones to certain generalised theta divisors contain secant varieties of the curve

X

. Here, we set

k = r

and deduce a similar statement for

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

using Theorem 5.6.…”

Section: Tangent Cones Of Higher Rank Brill–noether Locimentioning

confidence: 72%

“…Here, we recall briefly some essential facts, referring the reader to [6, 9] for details. The moduli space

U_{X} false(r, d false)

of stable bundles of rank

r

and degree

d

over

X

is an irreducible quasi‐projective variety of dimension

r^{2} false(g - 1 false) + 1

.…”

Section: Tangent Cones Of Higher Rank Brill–noether Locimentioning

confidence: 99%

“…If

E

is Petri

k

‐injective, then

Ker (\tilde{\cup})

is a vector subbundle of

{normalGr}^{k} \times H^{1} false(X, normalEnd E false)

. By [6, Theorem 2.4 (4)], the tangent cone

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

is, set‐theoretically,

⋃_{Λ \in {normalGr}^{k}} Im {(μ_{normalΛ})}^{⊥} = ⋃_{Λ \in {normalGr}^{k}} Ker false({\tilde{\cup}}_{Λ} false),

the second expression following from Serre duality as above. Projectivising, we obtain the following.…”

Section: Tangent Cones Of Higher Rank Brill–noether Locimentioning

confidence: 99%

“…It seems therefore natural to investigate what can be recovered from the tangent cones

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

at singular points

E

. Note that [6] 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%

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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 .

show abstract

“…By Theorem 2.2, for each such

normalΠ

there exists

Z_{normalΠ} \in {normalHilb}^{d} false(double-struckP E^{*} false)

such that

{scriptO}_{X} \otimes Π = {(E^{*})}_{Z_{Π}}

. Combining this observation with results from [6], we obtain the following generalisation of the Riemann–Kempf singularity theorem. In what follows, for any

ℓ ⩾ r

we denote by

U^{ℓ}

the open subset of

Gr (ℓ, H^{0} false(X, E false))

of subspaces which generically generate

E

.…”

Section: Introductionmentioning

confidence: 51%

“…In [6, § 5], it is shown that the tangent cones to certain generalised theta divisors contain secant varieties of the curve

X

. Here, we set

k = r

and deduce a similar statement for

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

using Theorem 5.6.…”

Section: Tangent Cones Of Higher Rank Brill–noether Locimentioning

confidence: 72%

“…Here, we recall briefly some essential facts, referring the reader to [6, 9] for details. The moduli space

U_{X} false(r, d false)

of stable bundles of rank

r

and degree

d

over

X

is an irreducible quasi‐projective variety of dimension

r^{2} false(g - 1 false) + 1

.…”

Section: Tangent Cones Of Higher Rank Brill–noether Locimentioning

confidence: 99%

“…If

E

is Petri

k

‐injective, then

Ker (\tilde{\cup})

is a vector subbundle of

{normalGr}^{k} \times H^{1} false(X, normalEnd E false)

. By [6, Theorem 2.4 (4)], the tangent cone

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

is, set‐theoretically,

⋃_{Λ \in {normalGr}^{k}} Im {(μ_{normalΛ})}^{⊥} = ⋃_{Λ \in {normalGr}^{k}} Ker false({\tilde{\cup}}_{Λ} false),

the second expression following from Serre duality as above. Projectivising, we obtain the following.…”

Section: Tangent Cones Of Higher Rank Brill–noether Locimentioning

confidence: 99%

“…It seems therefore natural to investigate what can be recovered from the tangent cones

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

at singular points

E

. Note that [6] 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%

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