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

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 $\left\{\frac{1}{r}(1,s_1),\frac{1}{r}(1,s_2),\dots ,\frac{1}{r}(1,s_k)\right\}$ for $k,r\in {\mathbb{Z}}_{>0}$, and $1\le s_i<r$ is coprime to $r$.

22 pages, 4 figures