Boundedness Results for Planar Linear Systems Assuming The Segre-Harbourne-Gimigliano-Hirschowitz Conjecture

TL;DR

This paper explores the implications of the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture on linear systems on blown-up projective planes, classifying systems by genus, self-intersection, and Cremona equivalence, with results on finiteness and minimality.

Contribution

It provides a classification of linear systems on blown-up projective planes assuming the conjecture, including finiteness results, minimal self-intersection characterization, and properties of base-point-free and birational systems.

Findings

Finiteness of linear systems with fixed genus up to Cremona equivalence.

Explicit computation of minimal self-intersection for given parameters.

Classification of systems with self-intersection ≤ 5.

Abstract

Let $X_n$ be the projective plane blown up at $n \geq 10$ general points. In this paper we give several consequences of the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture, that pertain to complete linear systems on $X_n$. We begin by classifying such systems $|C|$ with general irreducible member of genus $g \geq 2$ (up to Cremona equivalence), in terms of invariants of the adjoint systems $|C+mK|$. We then use this to prove that, for fixed $n \geq 10$ and $g\geq 2$, up to the action of the Cremona group, there exist finitely many complete linear systems on $X_n$ whose general member is irreducible of genus $g$. Further, there is a function $g\mapsto n(g)$ such that every such (effective) system is Cremona equivalent to a system in $X_{n(g)}$. The latter result is based on the explicit computation of the minimum possible self-intersection of an irreducible linear system with given $n$ and $\dim(|C|)$. We classify those systems which achieve the minimal self-intersection. We also classify the systems with $C^2 \leq 5$, whether or not they have minimal $C^2$ for the given $n$ and dimension. We finish by proving several statements concerning systems that are base-point-free, and systems that give birational maps to their image.