论文标题
超过有限生成的田地的平滑性曲面的线性家族
Linear families of smooth hypersurfaces over finitely generated fields
论文作者
论文摘要
令$ k $为有限生成的字段。我们构建了一个$ n $维线性系统$ \ MATHCAL {l} $ $ d $ $ d $ in $ \ mathbb {p}^n $在$ k $上定义的$ d $ in $ d $ n $ \ nathcal {l} $的每个成员在$ k $上定义的每个成员在特定的假设中都不是$ p $ dived（d div），当$ k $具有特征$ 0 $时，没有限制。此外，当$ p $ divies $ \ gcd（d，n+1）$时，我们将展出反例。
Let $K$ be a finitely generated field. We construct an $n$-dimensional linear system $\mathcal{L}$ of hypersurfaces of degree $d$ in $\mathbb{P}^n$ defined over $K$ such that each member of $\mathcal{L}$ defined over $K$ is smooth, under the hypothesis that the characteristic $p$ does not divide $\gcd(d, n+1)$ (in particular, there is no restriction when $K$ has characteristic $0$). Moreover, we exhibit a counterexample when $p$ divides $\gcd(d, n+1)$.
