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Multiplicity of the Saturated Special Fiber Ring of Height Three Gorenstein Ideals
Yairon Cid-Ruiz
,
Vivek Mukundan
Abstract
Let $R$ be a polynomial ring over a field and let $I\subset R$ 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 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 $I$.