Intersection Theory on Abelian-Quotient V-Surfaces and Q-Resolutions

Abstract

In this paper we study the intersection theory on surfaces with abelian quotient singularities and we derive properties of quotients of weighted projective planes. We also use this theory to study weighted blow-ups in order to construct embedded Q-resolutions of plane curve singularities and abstract Q-resolutions of surfaces.

Cartier and Weil divisors on varieties with quotient singularities

“…Then, it is possible to define the intersection number as the degree of the pull-back in the normalization of the compact divisor of the line bundle associated with the other divisor. These ideas will be developed in [1]. The following is an illustrative example.…”

“…For this definition, we need to describe abelian quotient singularities. In [1], we develop an intersection theory for surfaces with abelian quotient singularities, where the results presented in this paper are essential tools.…”

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

“…For instance, the behavior of the Lefschetz numbers and the zeta function of the monodromy in this setting was treated in [17] providing the corresponding A'Campo's formula [1]. Also, for plane curves, the local δ-invariant and explicit formulas for the self-intersections numbers of the exceptional divisors were calculated in [7] and [4] respectively.…”

“…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.…”