On a question of Dolgachev

TL;DR

This paper introduces a rational self-map on the space of plane curves of degree n, computes its degree for quartics, and addresses a question about the moduli space of genus 3 curves.

Contribution

It defines a new classical contravariant-based map for even degrees and determines its degree for quartics, solving Dolgachev's question.

Findings

The map has degree 15 for plane quartics.

It provides a new tool for studying moduli spaces of curves.

Answers a specific open question by Dolgachev.

Abstract

For each even, positive integer $n$, we define a rational self-map on the space of plane curves of degree $n$, using classical contravariants. In the case of plane quartics, we show that the degree of this map is 15. This answers a question of Dolgachev on the moduli space of curves of genus 3.