Formulas for the multiplicity of graded algebras (Q2841345)

Let \(A = A_0[A_1] \subseteq B = B_0[B_1]\) be a homogeneous inclusion of standard graded Noetherian rings with \(A_0 = B_0\) Artinian local. The author's main goal is to give a formula for the multiplicity of \(A\) in terms of the multiplicity of \(B\) and of local multiplicities of hyperplane sections along \(\text{Proj }(B)\). One of the main applications of this formula is to compute the multiplicity of special fiber rings; a special case of this construction gives homogeneous coordinate rings of Gauss images and of secant varieties. She also uses her formula to obtain the degree of the dual variety for any hypersurface, without any restriction on its singularities or on the dual variety (i.e. the dual variety does not need to be a hypersurface). In particular, she gives a generalization of Tessier's Plücker formula to hypersurfaces with non-isolated singularities. Her work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.