A geometric degree formula for A-discriminants and Euler obstructions of toric varieties

Abstract

We give explicit formulas for the dimensions and the degrees of A-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky [14]. Our formulas can be applied also to the case where the A-discriminant varieties are higher-codimensional and their degrees are described by the geometry of the configurations A. Moreover combinatorial formulas for the Euler obstructions of general (not necessarily normal) toric varieties will be also given.

Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

“…Under some weak additional conditions, a beautiful formula for the eigenvalues of the kth principal monodromy Φ n−k,0 was obtained by Oka [33,34] and Kirillov [20] etc. For related results, see also [11,26,27] etc. Moreover, the mixed Hodge structures of the Milnor fiber F 0 were precisely studied by Ebeling and Steenbrink [9] and Tanabe [45] etc.…”

Local Euler obstructions of toric varieties

“…This example is somewhat surprising, as it exhibits a variety with isolated singularities which has Euler-obstruction constantly equal to 1. Matsui and Takeuchi [21] shows that for normal and projective toric surfaces, the Eulerobstruction is constantly equal to 1 if and only if the variety is smooth. They conjectured the similar statement in higher dimensions.…”

“…For a normal toric surface X, we have that X is smooth if and only if Eu(X) = ½ X [21,Cor 5.7]. [21] conjecture that the corresponding statement should also hold for a higher dimensional normal and projective toric variety.…”

“…[21] conjecture that the corresponding statement should also hold for a higher dimensional normal and projective toric variety. As an application we present counterexamples to this conjecture.…”

“…The first algebraic formula was given by González-Sprinberg and Verdier [10]. Matsui and Takeuchi use a topological definition [21], which defines the local Euler obstruction of X inductively using the Whitney stratification of X. They use this definition to prove a formula for the local Euler obstruction on a (not necessarily normal) toric variety X.…”