Embedding Suzuki curves in $\mathbb{P}^4$

TL;DR

This paper investigates the projective embeddings of smooth models of plane Suzuki curves in four-dimensional projective space, analyzing their defining hypersurfaces and geometric properties.

Contribution

It provides explicit descriptions of hypersurfaces containing the curves and characterizes those of small degree, revealing limitations for higher-degree hypersurfaces.

Findings

Counted hypersurfaces of $ abla^4$ containing the curves

Provided geometric characterization of small-degree hypersurfaces

Proved the characterization does not extend to higher degrees

Abstract

Here we study the projective geometry of smooth models $X_n \subseteq \mathbb{P}^4$ of plane Suzuki curves $S_n$. The knowledge of a system of generators for the Weierstrass semigroup at the only singular point of the curve is shown to have relevant geometric consequences. In particular, here we explicitly count the hypersurfaces of $\mathbb{P}^4$ containing $X_n$ and provide a geometric characterization of those of small degree. We prove that the characterization cannot be extended to higher-degree hypersurfaces of $\mathbb{P}^4$.