Homogeneous algebraic varieties and transitivity degree (Q2099697)

Let \(X\) be an algebraic variety defined over an algebraically closed field \(\mathbb{K}\) of characteristic \(0\). If the automorphism group \({\mathrm{Aut}}(X)\) acts on \(X\) transitively, \(X\) is called a homogeneous variety. For any pairwise distinct points \(x_1, \ldots, x_m\in X\) and any pairwise distinct points \(y_1, \ldots, y_m\in X\), if there is an element \(\varphi \in {\mathrm{Aut}}(X)\) such that \(\varphi (x_i)=y_i\) for all \(1 \le i \le m\), the action of \({\mathrm{Aut}}(X)\) on \(X\) is called \(m\)-transitive. The transitivity degree \(\theta(X)\) is the maximal number such that the action is \(m\)-transitive. If the action is \(m\)-transitive for all positive integers \(m\), the action is called infinitely transitive and denoted by \(\theta(X)=\infty\). The authors show that if there is a nonconstant invertible regular function on an irreducible homogeneous variety \(X\), then \(\theta(X)=1\). They also show that if \(X\) is an irreducible homogeneous variety with \(\mathbb{K}[X]\neq \mathbb{K}\) and \(X\) is not quasi-affine, then \(\theta(X)=1\). An algebraic variety \(X\) is called a homogeneous space if there exists a transitive action of an algebraic group \(G\) on \(X\). In this case, \(X\) is identified with the left coset \(G/H\) where \(H\) is the stabilizer in \(G\) at a point of \(X\). By definition, any homogeneous space is a homogeneous variety, but not vice versa [\textit{I. V. Arzhantsev} et al., Sb. Math. 203, No. 7, 923--949 (2012; Zbl 1311.14059); translation from Mat. Sb. 203, No. 7, 3--30 (2012)]. By computing the transitivity degree of quasi-affine toric varieties and homogeneous spaces of linear algebraic groups with no nonconstant invertible functions, they propose the following conjecture: If \(X\) is an irreducible homogeneous quasi-affine variety with \(\dim X\ge 2\) and \(\mathbb{K}[X]^*=\mathbb{K}^*\), then \(\theta(X)=\infty\).