197 research outputs found
On mixed multiplicities of ideals
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.Comment: Minor corrections: a reference to a paper of B. Teissier added and
reference to results of B. Teissier and Rees-Sharp in the introduction
correcte
Mixed volume and an extension of intersection theory of divisors
Let K(X) be the collection of all non-zero finite dimensional subspaces of
rational functions on an n-dimensional irreducible variety X. For any n-tuple
L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the
number of solutions in X of a system of equations f_1 = ... = f_n = 0 where
each f_i is a generic function from the space L_i. In counting the solutions,
we neglect the solutions x at which all the functions in some space L_i vanish
as well as the solutions at which at least one function from some subspace L_i
has a pole. The collection K(X) is a commutative semigroup with respect to a
natural multiplication. The intersection index [L_1,..., L_n] can be extended
to the Grothendieck group of K(X). This gives an extension of the intersection
theory of divisors. The extended theory is applicable even to non-complete
varieties. We show that this intersection index enjoys all the main properties
of the mixed volume of convex bodies. Our paper is inspired by the
Bernstein-Kushnirenko theorem from the Newton polytope theory.Comment: 31 pages. To appear in Moscow Mathematical Journa
Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring
We combine work of Cox on the total coordinate ring of a toric variety and
results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and
monomial ideals to obtain a formula for computing the Euler characteristic of a
Weil divisor D on a complete simplicial toric variety in terms of graded pieces
of the Cox ring and Stanley-Reisner ring. The main point is to use Alexander
duality to pass from the toric irrelevant ideal, which appears in the
computation of the Euler characteristic of D, to the Stanley-Reisner ideal of
the fan, which is used in defining the Chow ring. The formula also follows from
work of Maclagan-Smith.Comment: 9 pages 1 figur
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