Ciliberto-Miranda degenerations of $\mathbb{CP}^2$ blown up in 10 points

TL;DR

This paper simplifies existing methods to construct degenerations of the complex projective plane blown up at multiple points, providing new lower bounds for Seshadri constants, notably achieving a bound of 117/370 for 10 points.

Contribution

It introduces a simplified approach to degenerations of blown-up projective planes, enabling more efficient bounds on Seshadri constants with fewer linear systems to check.

Findings

Established a new lower bound of 117/370 for 10-point Seshadri constant.

Simplified the construction method for degenerations of blown-up $ ext{CP}^2$.

Reduced computational complexity in verifying non-special linear systems.

Abstract

We simplify Ciliberto's and Miranda's method(arXiv:0812.0032) to construct degenerations of $\mathbb{CP}^2$ blown up in several points yielding lower bounds of the corresponding multi-point Seshadri constants. In particular we exploit an asymptotic result of the author (arXiv:math/0508561) which allows to check the non-specialty of much fewer linear systems on $\mathbb{CP}^2$. We obtain the lower bound 117/370 for the 10-point Seshadri constant on $\mathbb{CP}^2$.