Linear systems on rational surfaces

TL;DR

The paper extends the SHGH interpolation framework from $\mathbb P^2$ to Hirzebruch surfaces by formulating a SHGH-type conjecture on $\mathbb F_{e,r}$ and analyzing divisors in the form $D=aH_e+bF_e-\sum m_iE_i$ with associated virtual and expected dimensions. It establishes the equivalence of Conjecture 3 with Laface's conjecture and develops the necessary machinery to compare speciality via $(-1)$-curves and fixed components, including an algorithmic reduction to a fixed-component-free divisor. The main result proves the conjecture for $r\le e+4$ (with $e>0$), leveraging anticanonical surface properties and vanishing theorems to show non-speciality of nef divisors, while the corresponding part for $e=0$ recovers the classical SHGH case for $\mathbb P^2$. This work connects the Hirzebruch setting to established results by HJSA and JKS25, and contributes to broader questions on Seshadri constants and bounded negativity on rational surfaces.

Abstract

Motivated by various equivalent versions of the SHGH conjecture for $\mathbb{P}^2$ blown up at very general points, we propose a similar conjecture for Hirzebruch surfaces. We prove that this conjecture is true for the Hirzebruch surface $\mathbb{F}_e$ blown up at $r\leqslant e+4$ very general points.

Theorems & Definitions (27)

• Definition 1.2
• Definition 1.3
• Conjecture 1.4
• Theorem 1.5: Theorem \ref{['e+4']}
• Remark 1.6
• Conjecture 2.1
• Remark 2.3
• Definition 2.4
• Conjecture 2.5
• Lemma 2.6