Numerical Adjunction Formulas for Weighted Projective Planes and Lattice Points Counting

TL;DR

This paper derives explicit formulas for Ehrhart quasi-polynomials and polynomial space dimensions related to weighted projective planes, connecting lattice point counting with surface singularity invariants and providing a Numerical Adjunction Formula for singular curves.

Contribution

It introduces explicit formulas linking lattice point enumeration, surface quotient singularities, and polynomial spaces on weighted projective planes, offering new tools for algebraic geometry and combinatorics.

Findings

Explicit Ehrhart quasi-polynomial formulas for certain polyhedra.

A formula for the dimension of quasi-homogeneous polynomial spaces.

Interpretation as a Numerical Adjunction Formula for singular curves.

Abstract

This paper gives an explicit formula for the Ehrhart quasi-polynomial of certain 2-dimensional polyhedra in terms of invariants of surface quotient singularities. Also, a formula for the dimension of the space of quasi-homogeneous polynomials of a given degree is derived. This admits an interpretation as a Numerical Adjunction Formula for singular curves on the weighted projective plane.