Local Invariants on Quotient Singularities and a Genus Formula for Weighted Plane Curves

Abstract

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.

Numerical adjunction formulas for weighted projective planes and lattice point counting

“…Embedded Q-resolutions are a natural generalization of the usual embedded resolutions, for which some invariants, such as δ can be effectively calculated ( [14]). …”

“…Notice that the generic curve of degree kw is smooth (see [14,Lemma 5.4]). In that case, one has (c.f.…”

“…However, for a general d the generic curve of degree d in P 2 w is not necessarily quasi-smooth (see [14,Lemma 5.4]). The final goal of this paper is to revisit (5) …”

“…The quadratic polynomial g w,k = k(k − |w|) 2w + 1 in k is called the virtual genus (see [14,Definition 5.1]) and ∆ P (k) is a periodic function of periodw which is an invariant associated to the singularity P ∈ Sing(P 2 w ) (see Definition 3.8). The proof of Theorem 1.1 relies heavily on computations with Dedekind sums.…”

“…The next result aims to show that the previous combinatorial number ∆ P (k) has a geometric interpretation and can be computed via invariants of curve singularities on a singular surface. In order to do so we recall the recently defined invariant δ P (f ) of a curve ({f = 0}, P ) on a surface with quotient singularity (see [14,Section 4.2]) and we define a new invariant κ P (f ) in §3.1. Theorem 1.2.…”