Plane curves of fixed bidegree and their Aₖ-singularities

Abstract

We provide a tool how one can view a polynomial on the affine plane of bidegree \((a,b)\) -- by which we mean that its Newton polygon lies in the triangle spanned by \((a,0)\), \((0,b)\) and the origin -- as a curve in a Hirzebruch surface having nice geometric properties. As an application, we study maximal \(A_k\)-singularities of curves of bidegree \((3,b)\) and find the answer for \(b\leq 12\).