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

Let R be a polynomial ring over a field and let I ⊂ 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 .