标题： 包含在超曲面中的Veronese流形 显示英文标题

标题： Veronese varieties contained in hypersurfaces

摘要： 亚历克斯·沃尔德伦证明了对于射影 $n$-空间中足够一般的度数 $d$的超曲面，参数化包含在超曲面中的 $r$-维线性空间的Fano概形非空恰好在度数范围 $n\geq N_1(r,d)$内，其中“预期维数” $f_1(n,r,d)$非负，在这种情况下 $f_1(n,r,d)$等于（纯）维数。 利用Gleb Nenashev的工作，我们证明对于射影$n$空间中足够一般的度$d$超曲面，包含在超曲面中的$r$维$e$重Veronese流形的参数空间是非空的，且其纯维度等于在度范围$n\geq \widetilde{N}_e(r,d)$内的“预期维度”$f_e(n,r,d)$，该度范围是渐近精确的。 此外，我们证明对于$n\geq 1+N_1(r,d)$，参数化$r$维线性空间的 Fano 族是不可约的。 显示英文摘要

摘要： 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.