关于仿射空间的自同态与雅可比问题
On endomorphisms of affine spaces and the Jacobian problem
AI总结：
本文研究仿射空间平展自同态的纤维基数、像补集及几何次数，构造反例否证若干猜想，并证明雅可比猜想在特定复合条件下成立。
AI中文摘要：
设 $p$ 为素数。我们给出例子表明，在特征为 $p$ 的代数闭域 $k$ 上，仿射平面的平展自同态的纤维可以具有任意有限基数。设 $(l,m)\in\mathbb N\times\mathbb N^{\ast}$。我们给出这样的平展自同态的例子，其像的补集的基数为 $l$，且其几何次数为 $pm$。若干猜想被否证，特别是我们在 $k$ 上给出了 Kulikov 对 Bass 的“广义雅可比猜想”（针对曲面）的反例的类比。在每个 $k$ 上，Adjamagbo 的雅可比猜想的满射（分别地，满射且非满射）反例在维数 $2$（分别地，在所有维数至少 $3$）中被包含；例如，若 $p=2$，则我们证明对每个 $m$，在 $k$ 上维数至少 $3$ 的仿射空间存在几何次数为 $m$ 的满射平展自同态。若 $e:X\rightarrow X$ 是代数闭域 $K$ 上簇的自同态，则我们证明存在 $n\in\mathbb N$ 使得 $\Imm(e^n)=\Imm(e^{n+1})$，只要 (i) $e$ 是拟有限的，或 (ii) $\dim(X)\le 2$ 且 $X\setminus\Imm(e)$ 是有限的。我们给出在 $K$ 上维数至少 $3$ 的仿射平面的自同态的例子，其像的补集的基数为 $l$，且对所有 $n\in\mathbb N$ 有 $\Imm(e^n)\neq\Imm(e^{n+1})$。我们证明在 $K$ 上具有行列式为 $1$ 的雅可比矩阵的仿射空间的平展自同态的所有仿射模方案都是连通的，并分类所有光滑的此类模方案。我们证明在 $\mathbb C$ 上的雅可比猜想及其在 $k$ 上的 Adjamagbo 类比，对于仿射空间的平展自同态成立，这些自同态是复合 $g\circ f$，其中 $f$ 是拟有限的，并且在特定有限子集之外的余维数 $1$ 中是一个局部闭嵌入，而 $g$ 是省略一个坐标的投影。
英文摘要：
Let $p$ be a prime. We provide examples which show that étale endomorphisms of affine planes over an algebraically closed field $k$ of characteristic $p$ can have fibers of arbitrary finite cardinal. Let $(l,m)\in\mathbb N\times\mathbb N^{\ast}$. We provide examples of such étale endomorphisms whose images have complements of cardinality $l$ and whose geometric degrees are $pm$. Several conjectures are disproved, and in particular we provide an analog over $k$ of Kulikov's counterexample to the `Generalized Jacobian Conjecture' of Bass for surfaces. Surjective (resp.\ surjective and non-surjective) counterexamples to Adjamagbo's Jacobian Conjecture over each $k$ are included in dimension $2$ (resp.\ in all dimensions at least $3$); for instance, if $p=2$, then we show for each $m$ there exist surjective étale endomorphisms of the affine spaces over $k$ of dimension at least $3$ of geometric degree $m$. If $e:X\rightarrow X$ is an endomorphism of a variety over an algebraically closed field $K$, then we show that there exists $n\in\mathbb N$ such that $\Imm(e^n)=\Imm(e^{n+1})$ provided either (i) $e$ is quasi-finite or (ii) $\dim(X)\le 2$ and $X\setminus\Imm(e)$ is finite. We provide examples of endomorphisms of affine planes of dimensions at least $3$ over $K$ whose images have complements of cardinality $l$ and for all $n\in\mathbb N$ we have $\Imm(e^n)\neq\Imm(e^{n+1})$. We prove that all affine moduli schemes of étale endomorphisms of affine spaces with Jacobian matrices of determinant 1 over $K$ are connected and classify all those that are smooth. We prove that the Jacobian Conjecture over $\mathbb C$ and Adjamagbo's analog of it over $k$ hold for étale endomorphisms of affine spaces that are composites $g\circ f$, where $f$ is quasi-finite and a locally closed embedding in codimension $1$ outside a specific finite subset and $g$ is a projection that omits one coordinate.
