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

TL;DR

This work investigates the Wahl map for the normalization of a δ-nodal curve on a Hirzebruch surface. By blowing up the δ nodes and studying the ambient Gaussian map on the blow-up, the authors derive surjectivity criteria and compute the corank of the Wahl map for the normalization, showing it equals $h^{0}(\mathbb{F}_{n},\mathcal{O}_{\mathbb{F}_{n}}(-K_{\mathbb{F}_{n}}))$; this verifies Wahl's conjecture in this context. The approach hinges on a detailed analysis of logarithmic forms, higher direct images under blow-ups, and a strategic three-ample decomposition to establish vanishing on a blown-up product, yielding concrete embedding obstructions between different Hirzebruch surfaces. Overall, the results connect Wahl-type obstructions with the geometry of nodal curves on ruled surfaces and contribute to the understanding of non-surjectivity loci in moduli spaces of curves.

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 $n\neq m$.

Theorems & Definitions (57)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Proposition 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof
• Lemma 3.4