Multiplicity of the saturated special fiber ring of height three Gorenstein ideals

TL;DR

This paper derives a formula for the multiplicity of the saturated special fiber ring of certain Gorenstein ideals of height three, linking algebraic invariants to geometric properties of associated rational maps.

Contribution

It provides an explicit formula for the multiplicity of the saturated special fiber ring of Gorenstein ideals of height three, depending only on key algebraic parameters.

Findings

Derived a formula for the multiplicity based on variables, generators, and syzygies.

Connected multiplicity to the $j$-multiplicity and rational map properties.

Offered an effective method to analyze rational maps from minimal generators.

Abstract

Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.