Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon

TL;DR

This paper generalizes the winding number concept for lattice point sequences and applies it to classify certain Fano polygons based on their singularity content, providing new restrictions for their structure.

Contribution

It introduces a generalized winding number formula for lattice sequences and uses it to classify Fano polygons with specific singularity configurations.

Findings

Classified Fano polygons without T-singularities using winding number restrictions.

Derived conditions on residual singularities of Fano polygons.

Provided a new tool for analyzing the structure of Fano polygons.

Abstract

By generalising the notion of a unimodular sequence, we create an expression for the winding number of certain ordered sets of lattice points. Since the winding number of the vertices of a Fano polygon is necessarily one, we use this expression as a restriction to classify all Fano polygons without T-singularities and whose basket of residual singularities is of the form $\left\{ \frac{1}{r}(1,s_{1}), \frac{1}{r}(1,s_{2}), \ldots, \frac{1}{r}(1,s_{k}) \right\}$ for $k,r \in \mathbb{Z}_{>0}$, and $1 \leq s_{i} < r$ is coprime to $r$.