论文标题
具有密度理性点的四分之一和五重的高空曲面
Quartic and Quintic hypersurfaces with dense rational points
论文作者
论文摘要
令$ x_4 \ subset \ mathbb {p}^{n+1} $是无限字段$ k $上的dimension $ n \ geq 4 $的四分之一的高度表面。我们表明，如果任何一个$ x_4 $都包含一个线性子空间$尺寸的$ h \ geq \ geq \ geq \ geq \ geq \ geq \ {2，\ dim（λ\ cap \ cap \ text {sing}（x_4）） - 2 \} $或沿$ h $ y $ k $ $ k $ y-k $ k $ y-k $ y-s $ y-s $ y-s $ y-s $ y-k $ s $ y-k $ s $ y-s $ s $ y-k $ y-k $ s $ y-k $ y-k $ s pointion an $ k $。这改善了A. Prodonzan和J. Harris，B。Mazur，R。Pandharipande的四重奏。我们还为$ k $合理的$ 3 $ 3 $ folds的$ K $合理点提供了一个密度结果，并在数字字段上具有双平面，以及在$ C_R $字段上的五重奏超曲面的几个不合理性结果。
Let $X_4\subset\mathbb{P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field $k$. We show that if either $X_4$ contains a linear subspace $Λ$ of dimension $h\geq \max\{2,\dim(Λ\cap \text{Sing}(X_4))-2\}$ or has double points along a linear subspace of dimension $h\geq 3$, a smooth $k$-rational point and is otherwise general, then $X_4$ is unirational over $k$. This improves previous results by A. Predonzan and J. Harris, B. Mazur, R. Pandharipande for quartics. We also provide a density result for the $k$-rational points of quartic $3$-folds with a double plane over a number field, and several unirationality results for quintic hypersurfaces over a $C_r$ field.
