Weak Brill-Noether for rational surfaces. (English) Zbl 1401.14186
Budur, Nero (ed.) et al., Local and global methods in algebraic geometry. Conference in honor of Lawrence Ein’s 60th birthday, University of Illinois at Chicago, IL, USA, May 12–15, 2016. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3488-5/pbk; 978-1-4704-4850-9/ebook). Contemporary Mathematics 712, 81-104 (2018).
Summary: A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Göttsche and Hirschowitz prove that on \(\mathbb {P}^2\) every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if the first Chern class is nef.
For the entire collection see [Zbl 1397.14004].
For the entire collection see [Zbl 1397.14004].
MSC:
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14J26 | Rational and ruled surfaces |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
Keywords:
moduli spaces of sheaves; Brill-Noether theory; rational surfaces; Hirzebruch and del Pezzo surfacesReferences:
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