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 $δ$-nodal curve $C$ on a Hirzebruch surface $\mathbb{F}{n}$ for $n\geq 0$. Let $σ:X\rightarrow \mathbb{F}{n}$ be the blow up of $\mathbb{F}{n}$ along the $δ$ nodes of $C$ and let $\widetilde{C}$ be the normalization of $C$ under $σ$. Let $K{X}$ be the canonical bundle of $X$ and let $Ω^{1}_{X}$ be the sheaf of $1$-holomorphic forms on $X$. We give conditions for the surjectivity of the map $Φ{X,\mathcal{O}{X}(K_{X}+\widetilde{C})}: \bigwedge^{2}H^{0}(X,\mathcal{O}{X}(K{X}+\widetilde{C}))\rightarrow H^{0}(X,Ω^{1}{X}(2K{X}+2\widetilde{C}))$. Using this surjectivity, we analyze the Wahl map $Φ{\widetilde{C}}:\bigwedge^{2}H^{0}(\widetilde{C},Ω^{1}{\widetilde{C}})\rightarrow H^{0}(\widetilde{C},(Ω^{1}_{\widetilde{C}})^{\otimes 3})$ and compute the corank of $Φ{\widetilde{C}}$ in various cases. We prove that the corank of the Wahl map for the normalization of a $δ$-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 $δ$-nodal curve on a Hirzebruch surface $\mathbb{F}{n}$ cannot be embedded as $δ-$nodal curve on a different Hirzebruch surface $\mathbb{F}{m}$, for $n\neq m$.