Veronese varieties contained in hypersurfaces

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 $n\ge N_1(r,d)$ 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 $n\ge {\tilde{N}}_e(r,d)$ that is asymptotically sharp. Moreover, we show that for $n\ge 1+N_1(r,d)$, the Fano scheme parameterizing $r$-dimensional linear spaces is irreducible.

19 pages

References in corpus (2)

• A note on Fröberg's conjecture for forms of equal degrees
• Fano varieties and linear sections of hypersurfaces