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

Let R be a polynomial ring and let I subset of 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 Buse, 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.Let R be a polynomial ring and let I subset of 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 Buse, 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.A