On the Brill-Noether theory for K3 surfaces

Maxim Leyenson

Open Mathematics (2012)

  • Volume: 10, Issue: 4, page 1486-1540
  • ISSN: 2391-5455

Abstract

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Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c 1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in the sense of moduli is also generic in the sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). The harder part of the proof is proving the non-emptiness of the Brill-Noether loci. In the case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called the strong theorem of the Brill-Noether theory. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.

How to cite

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Maxim Leyenson. "On the Brill-Noether theory for K3 surfaces." Open Mathematics 10.4 (2012): 1486-1540. <http://eudml.org/doc/269408>.

@article{MaximLeyenson2012,
abstract = {Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c 1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in the sense of moduli is also generic in the sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). The harder part of the proof is proving the non-emptiness of the Brill-Noether loci. In the case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called the strong theorem of the Brill-Noether theory. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.},
author = {Maxim Leyenson},
journal = {Open Mathematics},
keywords = {Algebraic surfaces; K3 surfaces; Vector bundles; surfaces; correspondences between surfaces; moduli spaces of sheaves on surfaces; Brill-Noether loci},
language = {eng},
number = {4},
pages = {1486-1540},
title = {On the Brill-Noether theory for K3 surfaces},
url = {http://eudml.org/doc/269408},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Maxim Leyenson
TI - On the Brill-Noether theory for K3 surfaces
JO - Open Mathematics
PY - 2012
VL - 10
IS - 4
SP - 1486
EP - 1540
AB - Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c 1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in the sense of moduli is also generic in the sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). The harder part of the proof is proving the non-emptiness of the Brill-Noether loci. In the case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called the strong theorem of the Brill-Noether theory. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.
LA - eng
KW - Algebraic surfaces; K3 surfaces; Vector bundles; surfaces; correspondences between surfaces; moduli spaces of sheaves on surfaces; Brill-Noether loci
UR - http://eudml.org/doc/269408
ER -

References

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