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 $n\geq 0$. Let $\sigma:X\rightarrow \mathbb{F}_{n}$ be the blow up of $\mathbb{F}_{n}$ along the $\delta$ nodes of $C$ and let $\widetilde{C}$ be the normalization of $C$ under $\sigma$. Let $K_{X}$ be the canonical bundle of $X$ and let $\Omega^{1}_{X}$ be the sheaf of $1$-holomorphic forms on $X$. We give conditions for the surjectivity of the map $\Phi_{X,\mathcal{O}_{X}(K_{X}+\widetilde{C})}: \bigwedge^{2}H^{0}(X,\mathcal{O}_{X}(K_{X}+\widetilde{C}))\rightarrow H^{0}(X,\Omega^{1}_{X}(2K_{X}+2\widetilde{C}))$. Using this surjectivity, we analyze the Wahl map $\Phi_{\widetilde{C}}:\bigwedge^{2}H^{0}(\widetilde{C},\Omega^{1}_{\widetilde{C}})\rightarrow H^{0}(\widetilde{C},(\Omega^{1}_{\widetilde{C}})^{\otimes 3})$ and compute the corank of $\Phi_{\widetilde{C}}$ 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 $n\neq m$.