  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
$\mathbb{P}^4$.
