Curves on Hirzebruch Surfaces and Semistability

TL;DR

This work studies when a general pointed curve can interpolate through a prescribed number of general points on Hirzebruch surfaces by analyzing geometric Tevelev degrees. It shows that for $a\ge 2$ the interpolation counts vanish except in the exceptional case $a=2$ with $\beta= m[C_0]$, where the count matches the classical $\mathbb{P}^1$ problem under the dimension constraint $n=2m-g+1$. The authors develop a stability-based framework, employing the semistability of the restricted tangent bundle $f^{*}T_X$, the base-point-free pencil trick, and a lifting argument to relate $\mathcal{H}_2$-maps to degree $m$ maps to $\mathbb{P}^1$, establishing the equality ${\mathsf{Tev}}^{{\mathcal{H}}_2}_{g,n,m[C_0]}={\mathsf{Tev}}^{\mathbb{P}^1}_{g,n,m}$ with $n=2m-g+1$. These results connect interpolation on Hirzebruch surfaces to the well-understood $\mathbb{P}^1$ case and illustrate how semistability obstructions govern the existence and enumeration of interpolating maps; the paper also frames broader implications for stacks of maps to projective bundles and their stability properties.

Abstract

When $a\ge2$, we show that a general pointed curve never interpolates through the expected number of points in the Hirzebruch surface $\mathcal{H}_a$, with one exception. In the exceptional case, the number of such interpolating maps is determined by the geometric Tevelev degrees of $\mathbb{P}^1$, which have been previously computed.

Theorems & Definitions (18)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof