Multiplicity of the saturated special fiber ring of height two perfect ideals

TL;DR

This paper derives a formula for the multiplicity of the saturated special fiber ring of height two perfect ideals, linking it to elementary symmetric polynomials of syzygy degrees, and applies it to rational map properties.

Contribution

It provides a new explicit formula for the multiplicity of the saturated special fiber ring of certain perfect ideals, connecting algebraic invariants to elementary symmetric polynomials.

Findings

Formula for multiplicity involving elementary symmetric polynomial

Method to determine degree and birationality of rational maps

Calculation of j-multiplicity for these ideals

Abstract

Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.