[Brill-Noether et existence de faisceaux semi-stables pour les surfaces de del Pezzo]
Soit une surface del Pezzo de degré . Lorsque , on calcule la cohomologie d’un faisceau général dans , l’espace des modules des faisceaux semi-stables de Gieseker de caractère de Chern . Nous classons également les caractères de Chern pour lesquels le faisceau général dans 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 , 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.
Let be a del Pezzo surface of degree . When , we compute the cohomology of a general sheaf in , the moduli space of Gieseker semistable sheaves with Chern character . We also classify the Chern characters for which the general sheaf in 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 , we further show our construction of certain vector bundles implies the existence of stable and semistable sheaves with respect to the anti-canonical polarization.
Révisé le :
Accepté le :
Première publication :
Publié le :
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
CC-BY-ND 4.0
@article{AIF_2024__74_3_1189_0,
author = {Levine, Daniel and Zhang, Shizhuo},
title = {Brill{\textendash}Noether and existence of semistable sheaves for del {Pezzo} surfaces},
journal = {Annales de l'Institut Fourier},
pages = {1189--1227},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {74},
number = {3},
year = {2024},
doi = {10.5802/aif.3619},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3619/}
}
TY - JOUR AU - Levine, Daniel AU - Zhang, Shizhuo TI - Brill–Noether and existence of semistable sheaves for del Pezzo surfaces JO - Annales de l'Institut Fourier PY - 2024 SP - 1189 EP - 1227 VL - 74 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3619/ DO - 10.5802/aif.3619 LA - en ID - AIF_2024__74_3_1189_0 ER -
%0 Journal Article %A Levine, Daniel %A Zhang, Shizhuo %T Brill–Noether and existence of semistable sheaves for del Pezzo surfaces %J Annales de l'Institut Fourier %D 2024 %P 1189-1227 %V 74 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3619/ %R 10.5802/aif.3619 %G en %F AIF_2024__74_3_1189_0
Levine, Daniel; Zhang, Shizhuo. Brill–Noether and existence of semistable sheaves for del Pezzo surfaces. Annales de l'Institut Fourier, Tome 74 (2024) no. 3, pp. 1189-1227. doi : 10.5802/aif.3619. https://aif.centre-mersenne.org/articles/10.5802/aif.3619/
[1] Variations of Nagata’s conjecture, A celebration of algebraic geometry (Clay Mathematics Proceedings), Volume 18, American Mathematical Society, 2013, pp. 185-203 | MR | Zbl
[2] Degenerations of planar linear systems, J. Reine Angew. Math., Volume 501 (1998), pp. 191-220 | MR | Zbl
[3] Weak Brill–Noether for rational surfaces, Local and global methods in algebraic geometry (Contemporary Mathematics), Volume 712, American Mathematical Society, 2018, pp. 81-104 | DOI | MR | Zbl
[4] Brill–Noether theorems and globally generated vector bundles on Hirzebruch surfaces, Nagoya Math. J., Volume 238 (2020), pp. 1-36 | DOI | MR | Zbl
[5] Existence of semistable sheaves on Hirzebruch surfaces, Adv. Math., Volume 381 (2021), 107636, 96 pages | DOI | MR | Zbl
[6] Fibrés stables et fibrés exceptionnels sur , Ann. Sci. Éc. Norm. Supér., Volume 18 (1985) no. 2, pp. 193-243 | DOI | MR | Zbl
[7] Weak Brill–Noether for vector bundles on the projective plane, Algebraic geometry (Catania, 1993/Barcelona, 1994) (Lecture Notes in Pure and Applied Mathematics), Volume 200, Marcel Dekker, 1998, pp. 63-74 | MR | Zbl
[8] Complete linear systems on rational surfaces, Trans. Am. Math. Soc., Volume 289 (1985) no. 1, pp. 213-226 | DOI | MR | Zbl
[9] Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 2013
[10] Birational geometry of moduli spaces of sheaves and Bridgeland stability, Surveys on recent developments in algebraic geometry (Proceedings of Symposia in Pure Mathematics), Volume 95, American Mathematical Society (2017), pp. 101-148 | DOI | MR | Zbl
[11] The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, Cambridge University Press, 2010, xviii+325 pages | DOI | MR | Zbl
[12] Exceptional sheaves on Del Pezzo surfaces, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 58 (1994) no. 3, pp. 53-87 | DOI | MR | Zbl
[13] Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 39, Springer, 2000, xii+208 pages | DOI | MR | Zbl
[14] Cubic forms. Algebra, geometry, arithmetic, North-Holland Mathematical Library, 4, North-Holland, 1986, x+326 pages | MR | Zbl
[15] On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math., Volume 33 (1960/61), pp. 271-293 | DOI | MR | Zbl
[16] A description of Chern classes of semistable sheaves on a quadric surface, J. Reine Angew. Math., Volume 453 (1994), pp. 113-135 | DOI | MR | Zbl
[17] Versal families and the existence of stable sheaves on a del Pezzo surface, J. Math. Sci., Tokyo, Volume 3 (1996) no. 3, pp. 495-532 | MR | Zbl
[18] Components of the stack of torsion-free sheaves of rank on ruled surfaces, Math. Ann., Volume 301 (1995) no. 4, pp. 699-715 | DOI | MR | Zbl
[19] Irreducibility of moduli spaces of vector bundles on birationally ruled surfaces, Algebraic geometry (Catania, 1993/Barcelona, 1994) (Lecture Notes in Pure and Applied Mathematics), Volume 200, Marcel Dekker, 1998, pp. 201-211 | MR | Zbl
Cité par Sources :
