2025/10/27 by Dalbelo, Thaís M., Duarte, Daniel, Santos, Danilo da Nóbrega
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.23570
In this work we present a formula to compute the Euler characteristic of the Milnor fiber of non-degenerate functions f: X → ℂ with isolated critical set, where X is a 2-generic symmetric determinantal variety. The formula is obtained in two steps. Firstly, we explicitly describe the toric structure of those varieties. Secondly, we compute volumes of Newton polyhedra arising from the toric structure. The result then follows from Matsui-Takeuchi's formula for the Milnor fibers over toric varieties. As an application, we compute the local Euler obstruction of X at the origin. This, in turn, allow us to provide a family of odd-dimensional normal toric varieties with isolated singular point not satisfying Matsui-Takeuchi's conjecture of the characterization of smoothness through the local Euler obstruction.