Limit Weierstrass Points on the Kenyon-Smillie Family of Plane Quartics

TL;DR

This paper investigates the behavior of Weierstrass points in a specific family of plane quartic curves, especially as the curves degenerate into singular forms, providing explicit descriptions of these limit points.

Contribution

It explicitly determines the limit Weierstrass points on singular members of the Kenyon-Smillie family of plane quartics, extending understanding of degenerations in algebraic geometry.

Findings

Identified limit Weierstrass points on nodal quartic

Described limit points on union of line and cubic

Analyzed quadruple line case

Abstract

Costantini and Kappes gave an algebraic equation of the universal family over the Kenyon-Smillie (2,3,4)-Teichm\"uller curve. This equation gives rise to a family of projective plane quartic curves with three singular members. These singular curves are: (1) an integral nodal quartic, (2) the union of a line and a nodal cubic, and (3) a quadruple line. We determine the limit Weierstrass points on these singular curves.