Veronese varieties contained in hypersurfaces

TL;DR

This paper extends Waldron's results on linear spaces in hypersurfaces to Veronese varieties, establishing conditions for their existence, dimension, and irreducibility of the parameter spaces in high-degree hypersurfaces.

Contribution

It proves the existence and expected dimension of Veronese varieties in hypersurfaces for a sharp degree range, generalizing previous linear space results.

Findings

Veronese varieties exist in hypersurfaces within a sharp degree range.

Parameter spaces of these varieties have the expected dimension.

Fano schemes of linear spaces are irreducible for sufficiently large degrees.

Abstract

Alex Waldron proved that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the Fano scheme parameterizing $r$-dimensional linear spaces contained in the hypersurface is nonempty precisely for the degree range $n\geq N_1(r,d)$ where the "expected dimension" $f_1(n,r,d)$ is nonnegative, in which case $f_1(n,r,d)$ equals the (pure) dimension. Using work by Gleb Nenashev, we prove that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the parameter space of $r$-dimensional $e$-uple Veronese varieties contained in the hypersurface is nonempty of pure dimension equal to the "expected dimension" $f_e(n,r,d)$ in a degree range $n\geq \widetilde{N}_e(r,d)$ that is asymptotically sharp. Moreover, we show that for $n\geq 1+N_1(r,d)$, the Fano scheme parameterizing $r$-dimensional linear spaces is irreducible.