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 and embedding behavior.

Contribution

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

Findings

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

Hyperosculating points correspond exactly to fixed points of certain group elements.

Results improve understanding of the embedding and geometric structure of generalized Fermat curves.

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 hyperoscualting 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.