Fermat hypersurfaces and Subcanonical curves

TL;DR

This paper extends classical theorems to subcanonical curves, establishing a link between gonality, surfaces of minimal degree, and Fermat hypersurfaces, revealing new geometric characterizations.

Contribution

It generalizes the Enriques-Petri Theorem to s-subcanonical curves and connects Fermat hypersurfaces with these curves through apolarity.

Findings

s-subcanonical curves are (s+2)-gonal iff contained in minimal degree surfaces

Fermat hypersurfaces of degree s+2 are apolar to s-subcanonical (s+2)-gonal curves

establishes a new geometric correspondence between hypersurfaces and curves

Abstract

We extend the classical Enriques-Petri Theorem to $s$-subcanonical projectively normal curves, proving that such a curve is $(s+2)$-gonal if and only if it is contained in a surface of minimal degree. Moreover, we show that any Fermat hypersurface of degree $s+2$ is apolar to an $s$-subcanonical $(s+2)$-gonal projectively normal curve, and vice versa.