Lower semi-continuity of the Waldschmidt constants

TL;DR

This paper investigates the Waldschmidt constant for certain fat point schemes in the projective plane, proving its lower semi-continuity for up to 8 points and computing specific examples related to weak del Pezzo surfaces.

Contribution

It establishes the lower semi-continuity of the Waldschmidt constant for fat point schemes with up to 8 points and computes these constants for particular configurations.

Findings

Proved lower semi-continuity for up to 8 points

Calculated Waldschmidt constants for five-point schemes on weak del Pezzo surfaces

Extended understanding of Waldschmidt constants in algebraic geometry

Abstract

In this paper, we study the Waldschmidt constant of a generalized fat point subscheme $Z=m_1p_1+\cdots+m_rp_r$ of $\mathbb{P}^2$, where $p_1,\cdots,p_r$ are essentially distinct points on $\mathbb{P}^2$, satisfying the proximity inequalities. Furthermore, we prove its lower semi-continuity for $r\le 8$. Using this property, we also calculate the Waldschmidt constants of the fat point subschemes $Z=p_1+\cdots+p_5$ giving weak del Pezzo surfaces of degree 4.