Tropical curves of unibranch points and hypertangency

TL;DR

This paper investigates the properties of integral plane curves with a single unibranch point, establishing conditions relating local invariants and tropical curve isomorphisms, and deriving formulas for invariants and locus dimensions.

Contribution

It introduces equivalent numeric and geometric conditions for unibranch points on plane curves and provides explicit formulas for invariants and locus dimensions.

Findings

Local invariants are arithmetically related.

Tropical curves associated are isomorphic.

Formulas for delta-invariant and locus dimension.

Abstract

We study integral plane curves meeting at a single unibranch point and show that such curves must satisfy two equivalent conditions. A numeric condition: the local invariants of the curves at the contact point must be arithmetically related. A geometric condition: the tropical curves that we associate to the contact point must be isomorphic. Moreover, we prove closed formulas for the delta-invariant of a unibranch singularity, and for the dimension of the loci of curves with an assigned unibranch point. Our work is motivated by interest in the Lang exceptional set.