Fat point ideals in $\mathbb{K}[\mathbb{P}^N]$ with linear minimal free resolutions and their resurgences

Abstract

The resurgence of an ideal of fat points in $\mathbb{𝕂}\left[{\mathbb{P}}^N\right]$ with linear minimal free resolution can be expressed as the quotient of its initial degree and its Waldschmidt constant. This makes it possible to do computations of the resurgence of this type of ideal with less complexity than in general. In this paper, given a fat point subscheme of ${\mathbb{P}}^N$, we construct a new subscheme where its saturated ideal has a linear minimal free resolution. In particular, we show that the saturated ideal of a fat point subscheme $Z=\left(s-2\right)p_1+p_2+\cdots +p_s$, supported on $s\ge 3$ general points of ${\mathbb{P}}^2$, has a linear minimal free resolution, and we compute its resurgence.