The Poincare series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface

TL;DR

This paper introduces a Poincaré series for simple complete ideals on smooth surface local rings, unifying jumping numbers and multiplier ideal dimensions, and proves its rationality with an explicit formula.

Contribution

It defines a new algebraic invariant, the Poincaré series, and establishes its rationality, providing a concrete expression for simple complete ideals on smooth surfaces.

Findings

Poincaré series unifies jumping numbers and multiplier ideal dimensions.

Proves the Poincaré series is a rational function.

Provides an explicit formula for the Poincaré series.

Abstract

For a simple complete ideal $\wp$ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincar\'e series $P_{\wp}$, that gathers in an unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to $\wp$. This paper is devoted to prove that $P_{\wp}$ is a rational function giving an explicit expression for it.