# Rationality of Seshadri constants on general blow ups of $\mathbb{P}^2$

## TL;DR

This paper investigates the rationality of global Seshadri constants on blow-ups of the projective plane at very general points, introducing a submaximality threshold that predicts when these constants are rational or irrational.

## Contribution

It defines a submaximality threshold for each number of points blown up and conjectures its role in determining the rationality of Seshadri constants, extending the SHGH Conjecture.

## Key findings

- Definition of submaximality threshold for each blow-up case
- Conjecture linking submaximality threshold to Seshadri constant rationality
- Determination of thresholds assuming the conjecture holds

## Abstract

Let $X$ be a projective surface and let $L$ be an ample line bundle on $X$. The global Seshadri constant $\varepsilon(L)$ of $L$ is defined as the infimum of Seshadri constants $\varepsilon(L,x)$ as $x\in X$ varies. It is an interesting question to ask if $\varepsilon(L)$ is a rational number for any pair $(X, L)$. We study this question when $X$ is a blow up of $\mathbb{P}^2$ at $r \ge 0$ very general points and $L$ is an ample line bundle on $X$. For each $r$ we define a $\textit{submaximality threshold}$ which governs the rationality or irrationality of $\varepsilon(L)$. We state a conjecture which strengthens the SHGH Conjecture and assuming that this conjecture is true we determine the submaximality threshold.