On determinantal equations for curves and Frobenius split hypersurfaces

TL;DR

This paper investigates the existence of intrinsic determinantal equations for plane curves and Frobenius split hypersurfaces, establishing their existence in various positive characteristic cases and for special classes like Fano and Calabi-Yau hypersurfaces.

Contribution

It proves the existence of intrinsic determinantal equations for plane curves and hypersurfaces in positive characteristic, including Frobenius split and special classes like Fano and Calabi-Yau.

Findings

In characteristic two, all ordinary plane curves of genus at least one have intrinsic determinantal equations.

In characteristic three, all plane curves are intrinsic Pfaffians.

In any positive characteristic, plane curves are set-theoretically determinants of intrinsic matrices.

Abstract

I consider the problem of existence of intrinsic determinantal equations for plane projective curves and hypersurfaces in projective space and prove that in many cases of interest there exist intrinsic determinantal equations. In particular I prove (1) in characteristic two any ordinary, plane projective curve of genus at least one is given by an intrinsic determinantal equation (2) in characteristic three any plane projective curve is an intrinsic Pfaffian (3) in any positive characteristic any plane projective curve is set theoretically the determinant of an intrinsic matrix (4) in any positive characteristic, any Frobenius split hypersurface in ${\bf P}^n$ is given by set theoretically as the determinant of an intrinsic matrix with homogeneous entries of degree between $1$ and $n-1$. In particular this implies that any smooth, Fano hypersurface is set theoretically given by an intrinsic determinantal equation and the same is also true for any Frobenius split Calabi-Yau hypersurface.