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

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.