The Wahl map of the normalization of nodal curves on Hirzebruch surfaces

Abstract

In this paper we study the Wahl map for the normalization of a $\delta$-nodal curve $C$ on a Hirzebruch surface ${\mathbb{F}}_n$ for $n\ge 0$. Let $\sigma :X\to {\mathbb{F}}_n$ be the blow up of ${\mathbb{F}}_n$ along the $\delta$ nodes of $C$ and let $\tilde{C}$ be the normalization of $C$ under $\sigma$. Let $K_X$ be the canonical bundle of $X$ and let ${\Omega }_X^1$ be the sheaf of $1$-holomorphic forms on $X$. We give conditions for the surjectivity of the map ${\Phi }_{X,{\mathcal{O}}_X(K_X+\tilde{C})}:{\bigwedge }^2{H}^0(X,{\mathcal{O}}_X(K_X+\tilde{C}))\to H^0(X,{\Omega }_X^1(2K_X+2\tilde{C}))$. Using this surjectivity, we analyze the Wahl map ${\Phi }_{\tilde{C}}:{\bigwedge }^2{H}^0(\tilde{C},{\Omega }_{\tilde{C}}^1)\to H^0(\tilde{C},({\Omega }_{\tilde{C}}^1{)}^{\otimes 3})$ and compute the corank of ${\Phi }_{\tilde{C}}$ in various cases. We prove that the corank of the Wahl map for the normalization of a $\delta$-nodal curve on ${\mathbb{F}}_n$ is $h^0({\mathbb{F}}_n,{\mathcal{O}}_{{\mathbb{F}}_n}(-K_{{\mathbb{F}}_n}))$, that verifies a conjecture by Wahl. Furthermore, as an application of our results, we demonstrate that, under certain conditions, a $\delta$-nodal curve on a Hirzebruch surface ${\mathbb{F}}_n$ cannot be embedded as $\delta -$nodal curve on a different Hirzebruch surface ${\mathbb{F}}_m$, for $n\ne m$.

26 pages. Comments are welcome