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.
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.
We prove that if (C, 0) is a reduced curve germ on a rational surface singularity (X, 0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair C ⊂ X. Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann-Roch formula, valid for projective normal surfaces, introduced by Blache.
In this paper we extend the concept of Milnor fiber and Milnor number to curve germs on surface quotient singularities. A generalization of the local δ-invariant is defined and described in terms of a Q-resolution of the curve singularity. In particular, when applied to the classical case (the ambient space is a smooth surface) one obtains a formula for the classical δ-invariant in terms of a Q-resolution, which simplifies considerably effective computations. All these tools will finally allow for an explicit description of the genus formula of a curve defined on a weighted projective plane in terms of its degree and the local type of its singularities.
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