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New examples of theta divisors for some syzygy bundles

  • Abel CastorenaEmail author
  • Hugo Torres-López
Research Article
  • 6 Downloads

Abstract

Let C be a smooth complex irreducible projective curve of genus g, and let \((L,H^0(L))\) be a generated complete linear series of type \((d,r+1)\) over C. The syzygy bundle, denoted by \(M_L\), is the kernel of the evaluation map Open image in new window . The aim of this paper is twofold. Firstly, to give new examples of stable syzygy bundles admitting a theta divisor over Petri curves. We prove that if \(M_L\) is strictly semistable then \(M_L\) admits a theta divisor. Secondly, to study the cohomological semistability of \(M_L\), and in this direction we give another proof of cohomological semistability of \(M_L\) when L induces a birational map. This proof gives us precise conditions for the cohomological semistability of \(M_L\) where such conditions agree with the semistability conditions for \(M_L\). Finally, we relate these two properties by showing that under certain conditions on Petri curves the cohomological semistability of \(M_L\) implies the existence of reducible theta divisor for \(M_L\).

Keywords

Syzygy bundle Stability of vector bundles Theta divisor Cohomological stability of vector bundles 

Mathematics Subject Classification

14C20 14H10 14H51 14H60 

Notes

Acknowledgements

The second author warmly thanks the Centro de Ciencias Matemáticas (CCM, UNAM) in Morelia City for their hospitality and provided resources during his academic visit.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centro de Ciencias Matemáticas, UNAM Campus MoreliaMoreliaMexico
  2. 2.CONACyT-UAZ, Unidad Académica de MatemáticasUniversidad Autónoma de Zacatecas (UAZ)ZacatecasMexico

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