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Transactions of the American Mathematical Society

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Formulas for the multiplicity of graded algebras

Author: Yu Xie
Journal: Trans. Amer. Math. Soc. 364 (2012), 4085-4106
MSC (2010): Primary 13H15, 13A30; Secondary 14J70, 14B05
Published electronically: March 28, 2012
MathSciNet review: 2912446
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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 $\textrm {Proj} (B)$. 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.

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Additional Information

Yu Xie
Affiliation: Department of Mathematics, The University of Notre Dame, South Bend, Indiana 46556

Keywords: $j$-multiplicity, associated graded ring, special fiber ring, dual variety, PlĂĽcker formula
Received by editor(s): October 7, 2009
Received by editor(s) in revised form: July 26, 2010
Published electronically: March 28, 2012
Additional Notes: This paper is based on the author’s Ph.D. thesis, written under the direction of Professor Bernd Ulrich. The author sincerely thanks Professor Ulrich for suggesting the problem and for advice and many helpful discussions.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.