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 singularities in the basket of residual singularities all have equal Gorenstein index.