2013/10/29 by Kaveh, Kiumars, Khovanskii, A. G.
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary: 13H15 #Secondary: 11H06
paper · doi:10.48550/arxiv.1310.7979
Let R be the local ring of a point on a variety X over an algebraically closed field k. We make a connection between the notion of mixed (Samuel) multiplicity of m-primary ideals in R and intersection theory of subspaces of rational functions on X which deals with the number of solutions of systems of equations. From this we readily deduce several properties of mixed multiplicities. In particular, we prove a (reverse) Alexandrov-Fenchel inequality for mixed multiplicities due to Teissier and Rees-Sharp. As an application in convex geometry we obtain a proof of a (reverse) Alexandrov-Fenchel inequality for covolumes of convex bodies inscribed in a convex cone.