Jet schemes of toric surfaces

TL;DR

This paper analyzes the structure of jet schemes of normal toric surfaces, providing formulas for their components, associating a graph to their data, and relating these to the surface's resolution and analytical type.

Contribution

It determines the irreducible components of jet schemes of toric surfaces, introduces an invariant called index of speciality, and links the components to the surface's resolution.

Findings

Formulas for the number and dimensions of jet scheme components.

A weighted graph encoding the jet scheme data, equivalent to the surface's analytical type.

Classification of components via index of speciality and correspondence with exceptional divisors.

Abstract

For $m\in \mathbb{N}, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a normal toric surface $S.$ We give formulas for the number of these components and their dimensions. When $m$ varies, these components give rise to projective systems, to which we associate a weighted graph. We prove that the data of this graph is equivalent to the data of the analytical type of $S.$ Besides, we classify these irreducible components by an integer invariant that we call index of speciality. We prove that for $m$ large enough, the set of components with index of speciality $1,$ is in 1-1 correspondance with the set of exceptional divisors that appear on the minimal resolution of $S.$