Quot schemes, Segre invariants, and inflectional loci of scrolls over curves

Hitching, George H.

doi:10.1007/s10711-019-00463-z

Cited by 2 publications

(1 citation statement)

References 24 publications

(27 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof Suppose

ν \in P V

, and consider the elementary transformation

V_{Z}

with

Z

the single point

ν

. By [13, Corollary 3.4], (with

V = K_{X} M^{- 1} \otimes E

), the point

ψ (ν)

is the projectivised image of the coboundary map in

H^{0} false(X, V false) \to H^{0} false(X, V_{ν} false) \to H^{0} false(X, V_{ν} / V false) \to H^{1} false(X, V false)

and, similarly, the embedded tangent space to

ψ (P V)

ψ (ν)

is the projectivised image of

H^{0} (X, \frac{V_{ν} \otimes O_{X} (x)}{V}) \to H^{1} false(X, V false) .

Hence,

ψ

separates points if and only if for any distinct

ν, ν^{'} \in P V

the map

H^{0} false(X, V_{ν} / V false) \oplus H^{0} false(X, V_{ν^{'}} / V false) \to H^{1} false(X, V false)

…”

Section: Geometric Riemann–roch For Vector Bundlesmentioning

confidence: 99%

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

Hitching

2020

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

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

“…Proof Suppose

ν \in P V

, and consider the elementary transformation

V_{Z}

with

Z

the single point

ν

. By [13, Corollary 3.4], (with

V = K_{X} M^{- 1} \otimes E

), the point

ψ (ν)

is the projectivised image of the coboundary map in

H^{0} false(X, V false) \to H^{0} false(X, V_{ν} false) \to H^{0} false(X, V_{ν} / V false) \to H^{1} false(X, V false)

and, similarly, the embedded tangent space to

ψ (P V)

ψ (ν)

is the projectivised image of

H^{0} (X, \frac{V_{ν} \otimes O_{X} (x)}{V}) \to H^{1} false(X, V false) .

Hence,

ψ

separates points if and only if for any distinct

ν, ν^{'} \in P V

the map

H^{0} false(X, V_{ν} / V false) \oplus H^{0} false(X, V_{ν^{'}} / V false) \to H^{1} false(X, V false)

…”

Section: Geometric Riemann–roch For Vector Bundlesmentioning

confidence: 99%

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

Hitching

2020

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Secant loci of scrolls over curves

Hitching

2024

Moduli Spaces and Vector Bundles—New Trends

View full text Add to dashboard Cite

Given a curve C C and a linear series ℓ \ell on C C , the secant locus V e e − f ( ℓ ) V^{e-f}_e( \ell ) parametrises effective divisors of degree e e which impose at most e − f e-f conditions on ℓ \ell . For E → C E \to C a vector bundle of rank r r , we define determinantal subschemes H e e − f ( ℓ ) ⊆ H i l b e ( P E ) H^{e-f}_e ( \ell )\subseteq \mathrm {Hilb}^e ( \mathbb {P}E ) and Q e e − f ( V ) ⊆ Q u o t 0 , e ( E ∗ ) Q^{e-f}_e(V)\subseteq \mathrm {Quot}^{0, e} ( E^* ) which generalise V e e − f ( ℓ ) V^{e-f}_e( \ell ) , giving several examples. We describe the Zariski tangent spaces of Q e e − f ( V ) Q^{e-f}_e(V) , and give examples showing that smoothness of Q e e − f ( V ) Q^{e-f}_e(V) is not necessarily controlled by injectiveness of a Petri map. We generalise the Abel–Jacobi map and the notion of linear series to the context of Quot schemes. We give some sufficient conditions for nonemptiness of generalised secant loci, and a criterion in the complete case when f = 1 f = 1 in terms of the Segre invariant s 1 ( E ) s_1 (E) . This leads to a geometric characterisation of semistability similar to that in [Quot schemes, Segre invariants, and inflectional loci of scrolls over curves, Geom. Dedicata 205 (2020), 1–19]. Using these ideas, we also give a partial answer to a question of Lange on very ampleness of O P E ( 1 ) \mathcal {O}_{\mathbb {P}E}(1) , and show that for any curve, Q e e − 1 ( V ) Q^{e-1}_e(V) is either empty or of the expected dimension for sufficiently general E E and V V . When Q e e − 1 ( V ) Q^{e-1}_e(V) has and attains expected dimension zero, we use formulas of Oprea–Pandharipande and Stark to enumerate Q e e − 1 ( V ) Q^{e-1}_e(V) . We mention several possible avenues of further investigation.

show abstract

Quot schemes, Segre invariants, and inflectional loci of scrolls over curves

Cited by 2 publications

References 24 publications

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

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

Secant loci of scrolls over curves

Contact Info

Product

Resources

About