2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210467We give explicit formulas for the dimensions and the degrees of $A$-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky. 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.25 pages, Chapter 4 of the previous version was separated and became a part of a new paper submitted to the arxiv, revisedAlgebraic Geometry14M25, 14N05, 32S60, 33C70, 35A27text