On the Wahl map of curves with Brill-Noether-Petri condition
Let \(C\) be a curve contained on a \(K3\) surface \(S\); then the Wahl map for \(C\), \(\Psi_ C:\wedge^ 2H^ 0K_ C\to H^ 0K_ C^{\otimes 3}\) has a non-trivial cokernel. -- Assume \(Pic(S)=\mathbb{Z}\), spanned by the class of \(C\); then there is a rank 2 bundle on \(S\) which induces on \(C\) an extension \(0\to L\to E\to K_ C-L\to 0\) where \(L\) is a \(g^ 1_{s+1}\) on \(C\) \((g(C)=2s)\), such that the map \(H^ 0(K_ C-L)\to H^ 1(L)\) is 0. In this paper, the two previous facts are correlated for curves satisfying the Brill-Noether-Petri condition; the author shows that starting with an element of the cokernel of the Wahl map, one can define a linear series \(L=g^ 1_{s+1}\) on \(C\), with a nontrivial extension \(0\to L\to E\to K_ C-L\to 0\), the left map of which is surjective on global sections. -- The construction yields a new proof of the surjectivity of the Wahl map, for curves of genus \(g\geq 13\).
scientific article; zbMATH DE number 95206
scientific article