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

Abstract

Let R be a polynomial ring over a field and 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 [5], we get a formula for the jmultiplicity of I and an effective method to study a rational map determined by a minimal set of generators of I.

A study of nonlinear multiview varieties

“…, I p−1 are zero-dimensional and I p is either a height two perfect ideal or a height three Gorenstein ideal. In particular, this result generalizes [14, Theorem A] and [19,Theorem A] to the multigraded setting.…”

“…In this section, we consider some special families of rational maps for which we give exact formulas for the product of the degree of the rational map with the multidegrees of its image. We shall extend the computations obtained in [14,19] for perfect ideals of height two and Gorenstein ideals of height three. Throughout this section we continue using Setup 3.1.…”

“…for classical multiview varieties. In Section 5, we prove our third main result, Theorem 5.5, which computes the multidegrees of Y for a large family of rational maps extending previous work in [14,19] to the multigraded setting. In Section 6, we consider monomial rational maps G, and we show that in this case, the multidegrees of the image can be computed in terms of the mixed volumes of certain lattice polytopes.…”

“…In [14, Theorem A], this result is widely generalized using the saturated special fiber ring to the case of rational maps G : P r k P s k where the base locus is given by a perfect ideal of height two. The saturated special fiber ring has been recently used to successfully study rational maps in many different contexts, see [14,16,19,20]. It was introduced by Busé -Cid-Ruiz -D'Andrea [8] for studying the degree and birationality of rational maps.…”