The Waldschmidt constant of special fat flat subschemes in $\mathbb{P}^N$]{The Waldschmidt constant of special fat flat subschemes in $\mathbb{P}^N$

TL;DR

This paper constructs special fat flat subschemes in projective spaces, computes their Waldschmidt constants, and classifies certain non-reduced fat point subschemes in  with small Waldschmidt constants, revealing infinite families with prescribed constants.

Contribution

It introduces a new class of fat flat subschemes in  and higher, computes their Waldschmidt constants, and classifies non-reduced fat point subschemes in  with constants below 2.5.

Findings

Infinite families of fat flat subschemes with Waldschmidt constant d for any positive integer d.

Construction of fat flat subschemes with Waldschmidt constants equal to b/a for integers 1a<b.

Complete classification of non-reduced fat point subschemes in  with Waldschmidt constants less than 5/2.

Abstract

The purpose of this paper is to construct some special kind of subschemes in $\mathbb{P}^N$ with $ N\ge 3$, which we call them "fat flat subschemes" and compute their Waldschmidt constants. These subschemes are constructed by adding, in a particular way, a finite number of linear subspaces of $\mathbb{P}^N$ of many different dimensions to a star configuration in $\mathbb{P}^N$, with arbitrary preassigned multiplicities to each one of these linear subspaces, as well as the star configuration. Among other things, it will be shown that for every positive integer $d$, there are infinitely many fat flat subschemes in $\mathbb{P}^N$ with the Waldschmidt constant equal to $d$. In addition to this, for any two integers $1\le a<b$, we also construct a fat flat subscheme of the above type in some projective space $\mathbb{P}^M$, which its Waldschmidt constant is equal to $b/a$. In addition to these, all non-reduced fat points subschemes $Z$ in $\mathbb{P}^2$ with the Waldschmidt constants less than $5/2$ are classified.