Rank-two vector bundles on Halphen surfaces and the Gauss-Wahl map for du Val curves
[Fibrés vectoriels de rang 2 sur les surfaces de Halphen et application de Gauss-Wahl pour les courbes de du Val]
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 257-285.

Une courbe de du Val de genre g est une courbe plane de degré 3g ayant 8 points de multiplicité g, un point de multiplicité g-1 et pas d’autre singularité. Nous montrons que le corang de l’application de Gauss-Wahl pour une courbe de du Val générale de genre impair (>11) est égal à 1. Ceci, joint aux résultats de [1], montre que la caractérisation, obtenue dans [3], des courbes de Brill-Noether-Petri ayant une application de Gauss-Wahl non surjective comme sections hyperplanes de surfaces K3 et limites de celles-ci, est optimale.

A genus-g du Val curve is a degree-3g plane curve having 8 points of multiplicity g, one point of multiplicity g-1, and no other singularity. We prove that the corank of the Gauss-Wahl map of a general du Val curve of odd genus (>11) is equal to one. This, together with the results of [1], shows that the characterization of Brill-Noether-Petri curves with non-surjective Gauss-Wahl map as hyperplane sections of K3 surfaces and limits thereof, obtained in [3], is optimal.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.43
Classification : 14J28,  14H51
Mots clés : Courbes, surfaces K3, fibrés vectoriels
@article{JEP_2017__4__257_0,
     author = {Arbarello, Enrico and Bruno, Andrea},
     title = {Rank-two vector bundles on {Halphen} surfaces and the {Gauss-Wahl} map for du {Val} curves},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {257--285},
     publisher = {Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.43},
     zbl = {1368.14050},
     mrnumber = {3623355},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jep.43/}
}
TY  - JOUR
AU  - Arbarello, Enrico
AU  - Bruno, Andrea
TI  - Rank-two vector bundles on Halphen surfaces and the Gauss-Wahl map for du Val curves
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2017
DA  - 2017///
SP  - 257
EP  - 285
VL  - 4
PB  - Ecole polytechnique
UR  - http://www.numdam.org/articles/10.5802/jep.43/
UR  - https://zbmath.org/?q=an%3A1368.14050
UR  - https://www.ams.org/mathscinet-getitem?mr=3623355
UR  - https://doi.org/10.5802/jep.43
DO  - 10.5802/jep.43
LA  - en
ID  - JEP_2017__4__257_0
ER  - 
Arbarello, Enrico; Bruno, Andrea. Rank-two vector bundles on Halphen surfaces and the Gauss-Wahl map for du Val curves. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 257-285. doi : 10.5802/jep.43. http://www.numdam.org/articles/10.5802/jep.43/

[1] Arbarello, E.; Bruno, A.; Farkas, G.; Saccà, G. Explicit Brill-Noether-Petri general curves, Comment. Math. Helv., Volume 91 (2016) no. 3, pp. 477-491 | Article | MR 3541717 | Zbl 1354.14050

[2] Arbarello, E.; Bruno, A.; Sernesi, E. Mukai’s program for curves on a K3 surface, Algebraic Geom., Volume 1 (2014) no. 5, pp. 532-557 | Article | MR 3296804 | Zbl 1322.14062

[3] Arbarello, E.; Bruno, A.; Sernesi, E. On hyperplane sections of K3 surfaces, Algebraic Geom. (to appear) (arXiv:1507.05002) | Zbl 06849619

[4] Arbarello, E.; Saccà, G. Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties (2015) (arXiv:1505.00759) | Zbl 06863455

[5] Beauville, A. Le théorème de Torelli pour les surfaces K3: fin de la démonstration, Geometry of K3 surfaces: moduli and periods (Palaiseau, 1981/1982) (Astérisque), Volume 126, Société Mathématique de France, Paris, 1985, pp. 111-121 | Numdam | MR 785227 | Zbl 0577.14030

[6] Cantat, S.; Dolgachev, I. Rational surfaces with a large group of automorphisms, J. Amer. Math. Soc., Volume 25 (2012) no. 3, pp. 863-905 | Article | MR 2904576 | Zbl 1268.14011

[7] Eisenbud, D.; Koh, J.; Stillman, M. Determinantal equations for curves of high degree, Amer. J. Math., Volume 110 (1998) no. 3, pp. 513-539 (with an appendix with J. Harris) | Article | MR 944326

[8] de Fernex, T. On the Mori cone of blow-ups of the plane (2010) (arXiv:1001.5243)

[9] Franciosi, M.; Tenni, E. On Clifford’s theorem for singular curves, Proc. London Math. Soc. (3), Volume 108 (2014) no. 1, pp. 225-252 | Article | MR 3162826 | Zbl 1284.14044

[10] Green, M. L. Koszul cohomology and the geometry of projective varieties, J. Differential Geometry, Volume 19 (1984) no. 1, pp. 125-171 (with an appendix by M. Green and R. Lazarsfeld) | Article | MR 739785

[11] Huybrechts, D.; Lehn, M. The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010 | Article | Zbl 1206.14027

[12] Kulikov, V. S. Degenerations of K3 surfaces and Enriques surfaces, Izv. Akad. Nauk SSSR Ser. Mat., Volume 41 (1977) no. 5, pp. 1008-1042 | MR 506296 | Zbl 0387.14007

[13] Lange, H. Universal families of extensions, J. Algebra, Volume 83 (1983) no. 1, pp. 101-112 | Article | MR 710589 | Zbl 0518.14008

[14] Morrison, D. R. The Clemens-Schmid exact sequence and applications, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) (Ann. of Math. Stud.), Volume 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 101-119 | MR 756848 | Zbl 0576.32034

[15] Mukai, S. Curves and K3 surfaces of genus eleven, Moduli of vector bundles (Sanda, 1994; Kyoto, 1994) (Lecture Notes in Pure and Appl. Math.), Volume 179, Dekker, New York, 1996, pp. 189-197 | MR 1397987 | Zbl 0884.14010

[16] Nagata, M. On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A Math., Volume 33 (1960), pp. 271-293 | Article | MR 126444 | Zbl 0100.16801

[17] Persson, U.; Pinkham, H. Degeneration of surfaces with trivial canonical bundle, Ann. of Math. (2), Volume 113 (1981) no. 1, pp. 45-66 | Article | MR 604042 | Zbl 0426.14015

[18] Voisin, C. Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math., Volume 168 (1992) no. 3-4, pp. 249-272 | Article | MR 1161267 | Zbl 0767.14012

[19] Wahl, J. On cohomology of the square of an ideal sheaf, J. Algebraic Geom., Volume 6 (1997) no. 3, pp. 481-511 | MR 1487224 | Zbl 0892.14022

[20] Wahl, J. Hyperplane sections of Calabi-Yau varieties, J. reine angew. Math., Volume 544 (2002), pp. 39-59 | MR 1887888 | Zbl 1059.14053

Cité par Sources :