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.

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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 = {Enrico Arbarello and Andrea Bruno},
     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},
     mrnumber = {3623355},
     zbl = {1368.14050},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.43/}
}
Enrico Arbarello; Andrea Bruno. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.43/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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