Formulas for the multiplicity of graded algebras

TL;DR

This paper derives formulas expressing the multiplicity of graded algebras using local $j$-multiplicities, enabling calculations of multiplicities and degrees of dual varieties, generalizing previous results in algebraic geometry.

Contribution

It introduces new formulas for multiplicities of graded algebras based on local $j$-multiplicities, extending prior work to more general hypersurfaces and embeddings.

Findings

Formulas for multiplicity via local $j$-multiplicities.

Application to compute degrees of dual varieties.

Generalization of Teissier's Plücker formula.

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 $A\subseteq B$ 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 ${\rm Proj}\,(B)$. 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 Pl\"{u}cker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.