Plane curves of fixed bidegree and their $A_k$-singularities

TL;DR

This paper introduces a geometric tool to analyze bidegree (a,b) polynomials as curves on Hirzebruch surfaces and applies it to classify maximal $A_k$-singularities for certain bidegrees.

Contribution

It develops a method to interpret bidegree polynomials as curves on Hirzebruch surfaces and applies this to classify maximal $A_k$-singularities for bidegree (3,b) with b up to 12.

Findings

Classified maximal $A_k$-singularities for bidegree (3,b) with b ≤ 12.

Provided a geometric framework linking polynomials to Hirzebruch surfaces.

Enhanced understanding of singularities in algebraic curves of fixed bidegree.

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 $b\leq 12$.