Brill-Noether and existence of semistable sheaves for del Pezzo surfaces. (Brill-Noether et existence de faisceaux semi-stables pour les surfaces de del Pezzo.) (English. French summary) Zbl 07877210
Summary: Let \(X_m\) be a del Pezzo surface of degree \(9-m\). When \(m \leq 5\), we compute the cohomology of a general sheaf in \(M(\mathbf{v})\), the moduli space of Gieseker semistable sheaves with Chern character \(\mathbf{v}\). We also classify the Chern characters for which the general sheaf in \(M(\mathbf{v})\) 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 \(m\leq 6\), we further show our construction of certain vector bundles implies the existence of stable and semistable sheaves with respect to the anti-canonical polarization.
MSC:
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14J26 | Rational and ruled surfaces |
| 14J45 | Fano varieties |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
References:
| [1] | Ciliberto, Ciro; Harbourne, Brian; Miranda, Rick; Roé, Joaquim, A celebration of algebraic geometry, 18, Variations of Nagata’s conjecture, 185-203, 2013, American Mathematical Society · Zbl 1317.14017 |
| [2] | Ciliberto, Ciro; Miranda, Rick, Degenerations of planar linear systems, J. Reine Angew. Math., 501, 191-220, 1998 · Zbl 0943.14002 |
| [3] | Coskun, Izzet; Huizenga, Jack, Local and global methods in algebraic geometry, 712, Weak Brill-Noether for rational surfaces, 81-104, 2018, American Mathematical Society · Zbl 1401.14186 · doi:10.1090/conm/712/14343 |
| [4] | Coskun, Izzet; Huizenga, Jack, Brill-Noether theorems and globally generated vector bundles on Hirzebruch surfaces, Nagoya Math. J., 238, 1-36, 2020 · Zbl 1440.14202 · doi:10.1017/nmj.2018.17 |
| [5] | Coskun, Izzet; Huizenga, Jack, Existence of semistable sheaves on Hirzebruch surfaces, Adv. Math., 381, 96 p. pp., 2021 · Zbl 1466.14051 · doi:10.1016/j.aim.2021.107636 |
| [6] | Drezet, J.-M.; Le Potier, J., Fibrés stables et fibrés exceptionnels sur \({\bf{P}}_2\), Ann. Sci. Éc. Norm. Supér., 18, 2, 193-243, 1985 · Zbl 0586.14007 · doi:10.24033/asens.1489 |
| [7] | Göttsche, Lothar; Hirschowitz, André, Algebraic geometry (Catania, 1993/Barcelona, 1994), 200, Weak Brill-Noether for vector bundles on the projective plane, 63-74, 1998, Marcel Dekker · Zbl 0937.14010 |
| [8] | Harbourne, Brian, Complete linear systems on rational surfaces, Trans. Am. Math. Soc., 289, 1, 213-226, 1985 · Zbl 0609.14004 · doi:10.2307/1999697 |
| [9] | Hartshorne, Robin, Algebraic geometry, 52, 2013, Springer |
| [10] | Huizenga, Jack, Birational geometry of moduli spaces of sheaves and Bridgeland stability, Surveys on recent developments in algebraic geometry, 95, 101-148, 2017, American Mathematical Society · Zbl 1402.14056 · doi:10.1090/pspum/095/01639 |
| [11] | Huybrechts, Daniel; Lehn, Manfred, The geometry of moduli spaces of sheaves, xviii+325 p. pp., 2010, Cambridge University Press · Zbl 1206.14027 · doi:10.1017/CBO9780511711985 |
| [12] | Kuleshov, S. A.; Orlov, D. O., Exceptional sheaves on Del Pezzo surfaces, Izv. Ross. Akad. Nauk, Ser. Mat., 58, 3, 53-87, 1994 · Zbl 0842.14009 · doi:10.1070/IM1995v044n03ABEH001609 |
| [13] | Laumon, Gérard; Moret-Bailly, Laurent, Champs algébriques, 39, xii+208 p. pp., 2000, Springer · Zbl 0945.14005 · doi:10.1007/978-3-540-24899-6 |
| [14] | Manin, Yu. I., Cubic forms. Algebra, geometry, arithmetic, 4, x+326 p. pp., 1986, North-Holland · Zbl 0582.14010 |
| [15] | Nagata, Masayoshi, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math., 33, 271-293, 196061 · Zbl 0100.16801 · doi:10.1215/kjm/1250775912 |
| [16] | Rudakov, Alexei N., A description of Chern classes of semistable sheaves on a quadric surface, J. Reine Angew. Math., 453, 113-135, 1994 · Zbl 0805.14023 · doi:10.1515/crll.1994.453.113 |
| [17] | Rudakov, Alexei N., Versal families and the existence of stable sheaves on a del Pezzo surface, J. Math. Sci., Tokyo, 3, 3, 495-532, 1996 · Zbl 0883.14021 |
| [18] | Walter, Charles H., Components of the stack of torsion-free sheaves of rank \(2\) on ruled surfaces, Math. Ann., 301, 4, 699-715, 1995 · Zbl 0820.14029 · doi:10.1007/BF01446655 |
| [19] | Walter, Charles H., Algebraic geometry (Catania, 1993/Barcelona, 1994), 200, Irreducibility of moduli spaces of vector bundles on birationally ruled surfaces, 201-211, 1998, Marcel Dekker · Zbl 0947.14019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
