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]
Levine, Daniel1 ; Zhang, Shizhuo2
1 Department of Mathematics The Pennsylvania State University University Park PA 16802 (USA)
2 Max Planck Institute for Mathematics Vivatsgasse 7 53111 Bonn (Germany)
Annales de l'Institut Fourier, Online first, 39 p.

Soit X m une surface del Pezzo de degré 9-m. Lorsque m≤5, 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 m≤6, 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 X m be a del Pezzo surface of degree 9-m. When m≤5, 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 m≤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.

Reçu le :
Révisé le :
Accepté le :
Première publication :
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) 
@unpublished{AIF_0__0_0_A61_0,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},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3619},
     language = {en},
     note = {Online first},
}
TY  - UNPB
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
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3619
LA  - en
ID  - AIF_0__0_0_A61_0
ER  - 
%0 Unpublished Work
%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
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3619
%G en
%F AIF_0__0_0_A61_0
Levine, Daniel; Zhang, Shizhuo. Brill–Noether and existence of semistable sheaves for del Pezzo surfaces. Annales de l'Institut Fourier, Online first, 39 p.

[1] Ciliberto, Ciro; Harbourne, Brian; Miranda, Rick; Roé, Joaquim 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] Ciliberto, Ciro; Miranda, Rick Degenerations of planar linear systems, J. Reine Angew. Math., Volume 501 (1998), pp. 191-220 | MR | Zbl

[3] Coskun, Izzet; Huizenga, Jack 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] Coskun, Izzet; Huizenga, Jack Brill–Noether theorems and globally generated vector bundles on Hirzebruch surfaces, Nagoya Math. J., Volume 238 (2020), pp. 1-36 | DOI | MR | Zbl

[5] Coskun, Izzet; Huizenga, Jack Existence of semistable sheaves on Hirzebruch surfaces, Adv. Math., Volume 381 (2021), 107636, 96 pages | DOI | MR | Zbl

[6] Drezet, J.-M.; Le Potier, J. Fibrés stables et fibrés exceptionnels sur P 2 , Ann. Sci. Éc. Norm. Supér., Volume 18 (1985) no. 2, pp. 193-243 | DOI | MR | Zbl

[7] Göttsche, Lothar; Hirschowitz, André 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] Harbourne, Brian Complete linear systems on rational surfaces, Trans. Am. Math. Soc., Volume 289 (1985) no. 1, pp. 213-226 | DOI | MR | Zbl

[9] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 2013

[10] Huizenga, Jack 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] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, Cambridge University Press, 2010, xviii+325 pages | DOI | MR | Zbl

[12] Kuleshov, S. A.; Orlov, D. O. Exceptional sheaves on Del Pezzo surfaces, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 58 (1994) no. 3, pp. 53-87 | DOI | MR | Zbl

[13] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 39, Springer, 2000, xii+208 pages | DOI | MR | Zbl

[14] Manin, Yu. I. Cubic forms. Algebra, geometry, arithmetic, North-Holland Mathematical Library, 4, North-Holland, 1986, x+326 pages | MR | Zbl

[15] Nagata, Masayoshi On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math., Volume 33 (1960/61), pp. 271-293 | DOI | MR | Zbl

[16] Rudakov, Alexei N. 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] Rudakov, Alexei N. 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] Walter, Charles H. Components of the stack of torsion-free sheaves of rank 2 on ruled surfaces, Math. Ann., Volume 301 (1995) no. 4, pp. 699-715 | DOI | MR | Zbl

[19] Walter, Charles H. 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 :