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

Cid-Ruiz, Yairon

doi:10.1090/proc/14693

Multiplicity of the saturated special fiber ring of height two perfect ideals
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Proc. Amer. Math. Soc. 148 (2020), 59-70 Request permission

Abstract:

Let $R$ be a polynomial ring and let $I \subset R$ 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 by Busé, D’Andrea, and the author, 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 $I$.

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