Abstract
Let \(\mathbb K\) be an algebraically closed field of characteristic zero. We show that if the automorphisms group of a quasi-affine variety \(X\) over \(\mathbb K\) is infinite, then \(X\) is uniruled.
Highlights
Automorphism groups of open varieties have always attracted a lot of attention, but the nature of these groups is still not well-known
In [6], Iitaka proved that Aut(Y ) is finite if Y has a maximal logarithmic Kodaira dimension
We focus on the group of automorphisms of an affine or, more generally, quasi-affine variety over an algebraically closed field of characteristic zero
Summary
Automorphism groups of open varieties have always attracted a lot of attention, but the nature of these groups is still not well-known. We focus on the group of automorphisms of an affine or, more generally, quasi-affine variety over an algebraically closed field of characteristic zero. We prove the following: Theorem 1.1 Let X be a quasi-affine (in particular affine) variety over an algebraically closed field of characteristic zero. If X is a quasi-affine non-uniruled variety, the automorphism group Aut(X ) of X is finite. We show (cf Proposition 7.2) that for every k ≥ 1 and every finite group G there is a k-dimensional affine (smooth) non-uniruled variety. In this version our result is optimal.
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