Multiplicity of the Saturated Special Fiber Ring of Height Three Gorenstein Ideals

Vivek Mukundan,Yairon Cid-Ruiz

doi:10.1007/s40306-020-00410-1

Abstract

Let R be a polynomial ring over a field and let I ⊂ 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 Busé et al. (Proc. Lond. Math. Soc. 121(4):743–787, 2020) 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.

Highlights

The saturated special fiber ring is an algebra that was in introduced in [5] and that was initially motivated by the interest of studying rational maps with the use of syzygies and blow-up algebras
We extend the family of ideals for which the multiplicity of the saturated special fiber ring is known
Let I ⊂ R be a homogeneous Gorenstein ideal of height three, n = μ(I) be the minimal number of generators of I, and suppose that I is minimally generated by homogeneous polynomials of the same degree δ

Summary

Introduction

The saturated special fiber ring is an algebra that was in introduced in [5] and that was initially motivated by the interest of studying rational maps with the use of syzygies and blow-up algebras. (iv) I satisfies the condition Gd. the multiplicity of the saturated special fiber ring F(I) of I is given by n−d 2 e F(I) = Dd−1 i=0 n − 2 − 2i. There are some consequences of Theorem A It was introduced in [1] and serves as a generalization of the Hilbert-Samuel multiplicity for non m-primary ideals. In the second main application of Theorem A, we study rational maps that are determined by a homogeneous minimal set of generators of I. This result adds a new class of rational maps for which the syzygies of the base ideal determine the product of the degrees of the rational map and the corresponding image.

The multiplicity of the saturated special fiber ring

Some applications