Embedded Q-resolutions for Yomdin-Lê surface singularities

Abstract

In a previous work we have introduced and studied the notion of embedded Q-resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities. Here we explicitly compute an embedded Q-resolution of a Yomdin-Lê surface singularity (V, 0) in terms of a (global) embedded Q-resolution of their tangent cone by means of just weighted blow-ups at points. The generalized A'Campo's formula in this setting is applied so as to compute the characteristic polynomial. As a consequence, an exceptional divisor in the resolution of (V, 0), apart from the first one which might be special, contributes to its complex monodromy if and only if so does the corresponding divisor in the tangent cone. Thus the resolution obtained is optimal in the sense that the weights can be chosen so that every exceptional divisor in the Q-resolution of (V, 0), except perhaps the first one, contributes to its monodromy.

Numerical adjunction formulas for weighted projective planes and lattice point counting

“…V -manifolds and Quotient Singularities. We start giving some basic definitions and properties of V -manifolds, weighted projective spaces, embedded Qresolutions, and weighted blow-ups (for a detailed exposition see for instance [15,2,3,19,21]). Let us fix the notation and introduce several tools to calculate a special kind of embedded resolutions, called embedded Q-resolutions (see Definition 2.4), for which the ambient space is allowed to contain abelian quotient singularities.…”

“…Let us define the weighted blow-up π : X → X at a point P ∈ X with respect to w = (p, q). Since it will be used throughout the paper, we briefly describe the local equations of a weighted blow-up at a point P of type (d; a, b) (see [19,Chapter 1] for further details).…”

Cartier and Weil divisors on varieties with quotient singularities

“…projective V -manifolds also carry a Hodge structure and a natural notion of normal crossing divisors can be defined (called Q-normal crossing divisors). As it will be developed in the second author's Ph.D. thesis [10], the study of the so-called Q-resolutions (allowing quotient singularities, especially abelian) provides a better understanding of some families of singularities.…”

Semistable reduction of a normal crossing $$\mathbb {Q}$$ Q -divisor

“…Here we study the semistable reduction associated with an embedded Q-resolution so as to compute the mixed Hodge structure on the cohomology of the Milnor fiber in the isolated case using a generalization of Steenbrink's spectral sequence. Examples of Yomdin-Lê surface singularities are presented as an application.Let us sketch some definitions and properties about V -manifolds, weighted projective spaces, and weighted blow-ups, see [4,15] for a more detailed exposition.Definition 1.1. Let H = {f = 0} ⊂ C n+1 .…”

“…Let us sketch some definitions and properties about V -manifolds, weighted projective spaces, and weighted blow-ups, see [4,15] for a more detailed exposition.…”