On Stability of Tautological Bundles and their Total Transforms

Mistretta, Ernesto C.

doi:10.1007/s00032-019-00301-7

Cited by 6 publications

(4 citation statements)

References 9 publications

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“…Strategy of the Proofs. Our approach to proving Theorem 1.3, Theorem 1.5, and Theorem 1.6 is inspired by an approach that was very fruitful in the study of tautological bundles on symmetric products of curves [Mis19,Kru20,Kru23], and also on Hilbert schemes of points on surfaces [Sca09a, Sca09b, Sca20, Sta16, Kru14a, Kru14b, Kru18]: Translate everything to (equivariant) sheaves on the cartesian product, where the computations often become easier. In the curve case, this is simply done by the pull-back along the quotient morphism π : C (d) → C [d] .…”

Section: Summary Of Resultsmentioning

confidence: 99%

Extension groups of tautological bundles on symmetric products of curves

Krug

2022

Beitr Algebra Geom

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We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if $$E\ne \mathcal O_X$$ E ≠ O X is simple, then the natural map $${{\,\mathrm{\mathsf {Ext}}\,}}^1(E,E)\rightarrow {{\,\mathrm{\mathsf {Ext}}\,}}^1(E^{[n]},E^{[n]})$$ Ext 1 ( E , E ) → Ext 1 ( E [ n ] , E [ n ] ) is injective for every n. Along with previous results, this implies that $$E\mapsto E^{[n]}$$ E ↦ E [ n ] defines an embedding of the moduli space of stable bundles of slope $$\mu \notin [-1,n-1]$$ μ ∉ [ - 1 , n - 1 ] on the curve X into the moduli space of stable bundles on the symmetric product $$X^{(n)}$$ X ( n ) . The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on $$X^{(n)}$$ X ( n ) where the dimension of the tangent space jumps. We also prove that $$E^{[n]}$$ E [ n ] is simple if E is simple.

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Section: Summary Of Resultsmentioning

confidence: 99%

Extension groups of tautological bundles on symmetric products of curves

Krug

2022

Beitr Algebra Geom

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“…where dim X ¼ n. As codim X singðFÞ > 3, and E is a vector bundle, then by Lemma 2.1 we have: Restriction to smooth curves obtained in such a way is used in various ways in the literature: in some works of the author (cf. [7,8]), some complete intersection curves are constructed to prove the stability of vector bundles on higher dimensional varieties. Also, in [3], stability of Picard bundles is proven by restriction to curves which are intersection of theta divisors.…”

Section: Main Theoremmentioning

confidence: 99%

A remark on stability and restrictions of vector bundles to hypersurfaces

Mistretta

2021

Bol. Soc. Mat. Mex.

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“…Let C be a smooth projective curve over ℂ , and let L be a globally generated line bundle on C, with deg L = d and dim L = h 0 (C, L) − 1 , let V ⊆ H 0 (C, L) a subspace of dimension r + 1 generating L. Then M V,L ∶= ker(V ⊗ O C → L) is a rank r vector bundle, which appears in different ways and has been given different names in the literature (cf. [4,6,7,9,[11][12][13]).…”

Section: Introductionmentioning

confidence: 99%

On linear stability and syzygy stability

Castorena

Mistretta

Torres-López

2022

Abh. Math. Semin. Univ. Hambg.

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In previous works, the authors investigated the relationships between linear stability of a generated linear series |V| on a curve C, and slope stability of the syzygy vector bundle $$M_{V,L} := \ker (V \otimes \mathcal {O}_C \rightarrow L)$$ M V , L : = ker ( V ⊗ O C → L ) . In particular, the second named author and L. Stoppino conjecture that, for a complete linear system |L|, linear (semi)stability is equivalent to slope (semi)stability of $$M_{L}$$ M L . The first and third named authors proved that this conjecture holds in the two opposite cases: hyperelliptic and generic curves. In this work we provide a counterexample to this conjecture on any smooth plane curve of degree 7.

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On Stability of Tautological Bundles and their Total Transforms

Cited by 6 publications

References 9 publications

Extension groups of tautological bundles on symmetric products of curves

Extension groups of tautological bundles on symmetric products of curves

A remark on stability and restrictions of vector bundles to hypersurfaces

On linear stability and syzygy stability

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