Linear orbits of smooth plane curves

TL;DR

This paper studies the orbits of smooth plane curves under linear transformations, constructing resolutions and computing degrees based on geometric and automorphism properties.

Contribution

It introduces a method to resolve orbit closures and compute their degrees, linking these to curve degree, automorphisms, and flexes.

Findings

Degree of orbit closure depends on curve degree, automorphism group, and flexes.

Constructs explicit resolutions of orbit closures.

Provides formulas for degrees based on geometric invariants.

Abstract

The group PGL(3) of linear transformations of the projective plane acts naturally on the projective space parametrizing curves of a given degree. In this note we begin the study of the orbits of smooth curves under this action: we construct a resolution of the closure of the orbit of a given curve, and we use it to compute its degree. This turns out to depend only on the degree of the curve, the order of its automorphism group, and on the number and type of its flexes. This paper will appear on the Journal of Algebraic Geometry.