Cartier and Weil divisors on varieties with quotient singularities

Abstract

Abstract. The main goal of this paper is to show that the notions of Weil and Cartier Q-divisors coincide for V -varieties and give a procedure to express a rational Weil Divisor as a rational Cartier divisor. The theory is illustrated with weighted projective spaces and weighted blow-ups.

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.…”

Monodromy zeta function formula for embedded $\mathbf{Q}$-resolutions

“…The quotient of C n by a finite abelian group is always isomorphic to a quotient space of type (d; A), see [3] for a proof of this classical result. Different types (d; A) can give rise to isomorphic quotient spaces.…”

“…To do this, in [3], we proved that Cartier and Weil divisors agree on V -manifolds. This allows one to develop a rational intersection theory on varieties with quotient singularities and study weighted blow-ups at points, see [2].…”

Embedded Q-resolutions for Yomdin-Lê surface singularities

“…Let us sketch some definitions and properties about V -manifolds, weighted projective spaces, and weighted blow-ups, see [4,5,8] for a more detailed exposition. Also, the generalized A'Campo's formula for embedded Qresolutions is recalled, see [9].…”

“…In this paper, the new techniques developed in [4,5,9] are partially applied to study these two families of singularities. More precisely, we present here a detailed explicit description of an embedded Q-resolution for YLS in terms of a (global) embedded Q-resolution of their tangent cone.…”