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

TL;DR

This paper investigates the Wahl map for the normalization of nodal curves on Hirzebruch surfaces, providing conditions for surjectivity, computing the corank, and confirming a conjecture by Wahl.

Contribution

It establishes the corank of the Wahl map for normalized nodal curves on Hirzebruch surfaces and applies these results to curve embedding uniqueness.

Findings

Corank of the Wahl map equals the dimension of sections of K_{\u00f6}

Confirmed Wahl's conjecture on the corank for these curves

Demonstrated non-embeddability of certain nodal curves on different 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 $n\neq m$.