Desingularizations of some Weighted Projective Planes

TL;DR

This paper presents an algorithmic approach to desingularize certain weighted projective planes, specifically those with at least one trivial weight, by computing Hirzebruch-Jung continued fractions, and includes a computer implementation.

Contribution

It introduces an iterative method for Hirzebruch-Jung continued fraction decomposition and applies it to weighted projective planes with trivial weights, along with a computer program.

Findings

Effective desingularization algorithm for weighted projective planes

Successful implementation of a computer program for continued fraction computation

Application to specific cases of weighted projective planes

Abstract

In this paper we discuss the desingularization algorithm for a toric surface. In particular, we construct an iterable method of determining the Hirzebruch-Jung continued fraction decomposition. These results are then applied to weighted projective planes with at least one tivial weight, ${\mathbb P}(1,m,n)$. The paper concludes with the development of a computer program that computes this continued fraction decomposition.