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

Abstract

Let R R be a polynomial ring and let I ⊂ R 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 I . Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of I I . Applying ideas introduced by Busé, D’Andrea, and the author, we obtain the value of the j j -multiplicity of I I and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of I I .