Weierstrass weight of the hyperosculating points of generalized Fermat curves

TL;DR

This paper establishes an optimal lower bound for the Weierstrass weight of hyperosculating points on generalized Fermat curves, enhancing understanding of their geometric properties in the moduli space.

Contribution

It provides the first sharp lower bound for Weierstrass weights of hyperosculating points on generalized Fermat curves, applicable in a dense subset of their moduli space.

Findings

Lower bound for Weierstrass weight is optimal and sharp in a dense open set.

Identifies hyperosculating points as fixed points of group elements.

Connects geometric properties with moduli space structure.

Abstract

Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F\subset S$ is the set of fixed points of the non-trivial elements of the group $H$, then $F$ is exactly the set of hyperosculating points of the standard embedding $S\hookrightarrow {\mathbb{P}}^{n}$. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.