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Acta Mathematica Vietnamica

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

icon-email Yairon Cid-Ruiz , Vivek Mukundan

Abstract

Let R be a polynomial ring over a field and let IR 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.