Brill–Noether theory on Hirzebruch surfaces

Costa, Laura; Miró‐Roig, Rosa M.

doi:10.1016/j.jpaa.2009.12.006

Cited by 5 publications

(1 citation statement)

References 17 publications

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“…REMARK 5.4. From [5,Proposition 2.4] whenever c 2 0 the moduli space M X ,H (2, c 1 , c 2 ) is a non-empty generically smooth, irreducible, quasi-projective variety of the expected dimension dim(M X ,H (2,…”

Section: Emptiness Of Brill-noether Loci Of Rank Two Stable Vector Bu...mentioning

confidence: 99%

A Note on Rank Two Stable Bundles Over Surfaces

Reyes-Ahumada¹,

Roa-Leguizamón²,

Torres-López³

2021

Glasgow Math. J.

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Abstract. Let π : X → C be a fibration with integral fibers over a curve C and consider a polarization H on the surface X. Let E be a stable vector bundle of rank 2 on C. We prove that the pullback π*(E) is a H-stable bundle over X. This result allows us to relate the corresponding moduli spaces of stable bundles $${{\mathcal M}_C}(2,d)$$ and $${{\mathcal M}_{X,H}}(2,df,0)$$ through an injective morphism. We study the induced morphism at the level of Brill–Noether loci to construct examples of Brill–Noether loci on fibered surfaces. Results concerning the emptiness of Brill–Noether loci follow as a consequence of a generalization of Clifford’s Theorem for rank two bundles on surfaces.

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Section: Emptiness Of Brill-noether Loci Of Rank Two Stable Vector Bu...mentioning

confidence: 99%

A Note on Rank Two Stable Bundles Over Surfaces

Reyes-Ahumada¹,

Roa-Leguizamón²,

Torres-López³

2021

Glasgow Math. J.

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Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces

Costa,

Macías Tarrío

2024

Mediterr. J. Math.

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Let X be a ruled surface over a nonsingular curve C of genus $$g\ge 0.$$ g ≥ 0 . Let $$M_H:=M_{X,H}(2;c_1,c_2)$$ M H : = M X , H ( 2 ; c 1 , c 2 ) be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes $$c_i:=c_i(E)$$ c i : = c i ( E ) for $$i=1,2.$$ i = 1 , 2 . The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space $$M_H$$ M H in terms of its Brill–Noether locus $$W_H^k(2;c_1,c_2),$$ W H k ( 2 ; c 1 , c 2 ) , whose points correspond to stable vector bundles in $$M_H$$ M H having at least k independent sections. We deal with the non-emptiness of this Brill–Noether locus, getting in most of the cases sharp bounds for the values of k such that $$W_H^k(2;c_1,c_2)$$ W H k ( 2 ; c 1 , c 2 ) is non-empty.

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On translated rank-2 Brill-Noether loci on regular surfaces

Filimon

2022

Arch. Math.

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Brill–Noether theory on Hirzebruch surfaces

Cited by 5 publications

References 17 publications

A Note on Rank Two Stable Bundles Over Surfaces

A Note on Rank Two Stable Bundles Over Surfaces

Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces

On translated rank-2 Brill-Noether loci on regular surfaces

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