Rank-two vector bundles on Halphen surfaces and the Gauss-Wahl map for du Val curves

Arbarello, Enrico; Bruno, Andrea

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.

Classification : 14J28, 14H51
Keywords: Curves, K3 surfaces, vector bundles
Mot clés : Courbes, surfaces K3, fibrés vectoriels

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