A note on the \(G\)-Sarkisov program (Q2032789)

A pair \((Z,\Phi)\) is a data of a normal variety \(Z\) and a \(\mathbb{Q}\)-divisor \(\Phi\) such that \(K_Z+\Phi\) is \(\mathbb{Q}\)-Cartier where \(K_Z\) is the canonical divisor attached to \(Z\). A pair is said to be Kawamata log terminal or klt, if for any birational morphism \(\mu:\tilde{Z}\rightarrow Z\) with \(\tilde{Z}\) smooth, we have \(K_{\tilde{Z}}=\mu^*(K_Z+\Phi)+\sum a_EE\) with \(a_E>-1\) for all \(E\). A variety \(Z\) is said to be terminal if for any birational morphism \(\mu:\tilde{Z}\rightarrow Z\) with \(\tilde{Z}\) smooth we have \(K_{\tilde{Z}}=\mu^*K_Z+\sum a_EE\) with \(a_E>0\) for all \(E\subseteq \mathrm{Exc}(\mu)\). For a connected algebraic group \(G\) a pair \((Z,\Phi)\) is said to be a \(G\)-pair if \(G\) acts on \(Z\) regularly and for all \(g\in G, g.\Phi=\Phi\). For a normal projective and \(\mathbb{Q}\)-factorial variety \(Z\), a corner stone result describing the geometric properties of \(Z\) by the behaviour of \(K_Z\) is the Cone and Contraction theorem due to Mori. It describes the curves which have negative intersection with \(K_Z\) and says that they can be contracted. The \(K_Z\)-minimal model program, or \(K_Z\)-MMP, is a sequence of elementary birational maps \(Z\rightarrow Z_1\rightarrow Z_2\rightarrow \cdots\) which are elementary, in the sense that they are defined using the simplest contractions given by the Cone and Contraction theorem, called extremal contractions. The process is expected to end and there are two possible outcomes depending on the geometry of \(Z\). Either the outcome is a variety \(X\) with nef canonical bundle, that is, such that \(K_X.C\geq 0\) for every curve \(C\subseteq X\); or \(X\) is a Mori fibre space, that is, there is a fibration \(\phi:X\rightarrow T\) such that \(\rho(X)=\rho(T)+1\) and \(-K_X\) restricted to the fibres of \(\phi\) is ample. The Cone and Contraction theorem holds for a klt pair and we can run the \(K_Z+\Phi\)-MMP. In this case, it is still an open problem whether the \(K_Z+\Phi\)-MMP stops in general, even when \(\Phi=0\), but there is an important class of varieties for which it is known to terminate, the ones which are covered by curves which have negative intersection with \(K_Z+\Phi\). Varieties with this property are uniruled. In this case, after a finite number of steps, we find a Mori fibre space which is not uniquely determined. The Sarkisov program describes the relation between two different Mori fibre spaces that are outcomes of two MMP on the same variety. The author proves the following result, that two Mori fibre spaces that are outcomes of two G-equivariant MMP on the same \(G\)-pair are related by a sequence of \(G\)-equivariant Sarkisov links (defined in the article) following the proof of the Sarkisov program by \textit{C. D. Hacon} and \textit{J. McKernan} [J. Algebr. Geom. 22, No. 2, 389--405 (2013; Zbl 1267.14024)]. Let \(G\) be a connected algebraic group. Let \((Z,\Phi)\) be a klt pair such that \(G<\Aut^0(Z)\) and \(G\) leaves \(\Phi\) invariant. Let \(\Phi:X\rightarrow S\) and \(\psi: Y\rightarrow T\) be two Mori fibre spaces obtained from \((Z,\Phi)\) via a \(G\)-equivariant MMP. Then \(X\) and \(Y\) are related by a sequence of \(G\)-equivariant Sarsikov links and in every such link the horizontal dotted arrows are compositions of \(G\)-equivariant flops with respect to a suitable boundary. As a consequence, the author obtains the following characterisation of subgroups of \(\mathrm{Bir}(W)\) that are maximal among the connected groups acting rationally on \(W\). Let \(W\) be an uniruled variety and let \(G\) be a connected algebraic group acting rationally on \(W\). Then \(G\) is maximal among the connected groups acting rationally on \(W\) if and only if \(G=\Aut^0(X)\) where \(\phi:X\rightarrow S\) is a Mori fibre space and for every Mori fibre space \(\psi:Y\rightarrow T\) which is related of \(\phi\) by a finite sequence of \(G\)-Sarkisov links we have \(G=\Aut^0(Y)\).