AI 中文总结
研究射影空间自同态临界轨迹，通过利用多项式环基本事实及引入特定多项式，证明次数为\(d\)的自同态临界概型是整超曲面，推广相关结果到任意特征。
AI 中文摘要
设\(K\)为代数闭域，且\(n,d\geq2\)。我们证明了\(\mathbb{P}^n_K\)上一般次数为\(d\)的自同态的临界概型是一个整超曲面。这将英格拉姆 - 拉马达斯 - 西尔弗曼的结果推广到任意特征。我们利用多项式环的基本事实表明某些精心选择的例子具有绝对不可约雅可比行列式。为处理\(K\)的特征整除\(d\)的狂野情形，我们引入一个模仿齐次雅可比行列式的多项式。
英文摘要
Let $K$ be an algebraically closed field and let $n, d \geq 2$. We show that the critical scheme of a general endomorphism of $\mathbb{P}^n_K$ of degree $d$ is an integral hypersurface. This extends a result of Ingram--Ramadas--Silverman to arbitrary characteristic. We use basic facts about polynomial rings to show that certain carefully chosen examples have absolutely irreducible Jacobian. To handle the wild case where the characteristic of $K$ divides $d$, we introduce a polynomial that imitates the homogeneous Jacobian determinant.
Comments32 pages
