Formulas for the multiplicity of graded algebras

Abstract

Let R be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of R by means of local j-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion A ⊆ B of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of A in terms of that of B and of local j-multiplicities of hyperplane sections along Proj (B). Our formulas can be used to find the multiplicity of special fiber rings and to obtain the degree of dual varieties for any hypersurface. In particular, it gives a generalization of Teissier's Plücker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.

Multiplicities of classical varieties

“…Related formulas expressing e F (I) in terms of multiplicities of I can be found in [49,Section 6] and [51,Section 5]. The assumption[ I s ) sat ] r = [I s ] r in (iii) is rather strong, but essential for the theorem as observed in the following example.…”

Relative multiplicities of graded algebras

“…It gives a lower bound for the difference of the multiplicities of A and B. Xie [14] has recently generalized the above result by giving the extra terms required to make it an equality.…”

Blowups and Fibers of Morphisms

“…Results of this type were obtained before by Simis et al. [49], Validashti [52], Xie [53] and by Jeffries et al. [32].…”

“…Moreover, we are able to relate the degree of the map Ψ, the degree of the image, and the j-multiplicity of the ideal I. Results of this type were obtained before by Simis, Ulrich and Vasconcelos [49], Validashti [52], Xie [53] and by Jeffries, Montano, and Varbaro [32].…”

“…According to [53, 2.5] (see also [1, 3.8] and [42, 3.6]) in the language of Corollary 3.7. On the other hand, is regular on , and is a homogeneous element of of degree ; so, and the assertion follows from Corollary 3.7.…”