The characteristic numbers of quartic plane curves

TL;DR

This paper computes the characteristic numbers of smooth plane quartic curves using intersection theory, confirming Zeuthen's classical predictions, and also provides a brief calculation for plane cubics as an example.

Contribution

It introduces a method to compute characteristic numbers of quartic plane curves via intersection theory on moduli spaces, completing Zeuthen's predictions.

Findings

Characteristic numbers of smooth plane quartics are explicitly computed.

Verification of Zeuthen's classical predictions for these curves.

A supplementary calculation for plane cubics is provided.

Abstract

The characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen's prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.