Brill–Noether and existence of semistable sheaves for del Pezzo surfaces
Annales de l'Institut Fourier, Online first, 39 p.

Let X m be a del Pezzo surface of degree 9-m. When m5, we compute the cohomology of a general sheaf in M(v), the moduli space of Gieseker semistable sheaves with Chern character v. We also classify the Chern characters for which the general sheaf in M(v) is non-special, i.e. has at most one nonzero cohomology group. Our results hold for arbitrary polarizations, slope semistability, and semi-exceptional moduli spaces. When m6, we further show our construction of certain vector bundles implies the existence of stable and semistable sheaves with respect to the anti-canonical polarization.

Soit X m une surface del Pezzo de degré 9-m. Lorsque m5, on calcule la cohomologie d’un faisceau général dans M(v), l’espace des modules des faisceaux semi-stables de Gieseker de caractère de Chern v. Nous classons également les caractères de Chern pour lesquels le faisceau général dans M(v) est non spécial, c’est-à-dire a au plus un groupe de cohomologie non nul. Nos résultats sont valables pour les polarisations arbitraires, la semi-stabilité des pentes et les espaces de modules semi-exceptionnels. Lorsque m6, nous montrons en outre que notre construction de certains fibrés vectoriels implique l’existence de faisceaux stables et semi-stables par rapport à la polarisation anti-canonique.

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DOI: 10.5802/aif.3619
Classification: 14J60, 14J26, 14J45, 14D20, 14F05
Keywords: Moduli spaces of sheaves, del Pezzo surfaces, Brill–Noether theory, Bogomolov inequalities.
Mot clés : Espace des modules des faisceaux, surfaces del Pezzo, théorie de Brill–Noether, inégalités de Bogomolov.
Levine, Daniel 1; Zhang, Shizhuo 2

1 Department of Mathematics The Pennsylvania State University University Park PA 16802 (USA)
2 Max Planck Institute for Mathematics Vivatsgasse 7 53111 Bonn (Germany)
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Levine, Daniel; Zhang, Shizhuo. Brill–Noether and existence of semistable sheaves for del Pezzo surfaces. Annales de l'Institut Fourier, Online first, 39 p.

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