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

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.