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

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 $\mathbb{P}^4$.