Prokhorov, Yuri; Shramov, Constantin \(p\)-subgroups in the space Cremona group. (English) Zbl 1423.14099 Math. Nachr. 291, No. 8-9, 1374-1389 (2018). Serre asked whether finite \(p\)-groups in \(\mathrm{Cr}_r=\mathrm{Bir}(\mathbb{P}^5)\) are abelian and generated by as most \(r\) elements, if \(p\) is a prime number large enough (dependent on \(r\)). Serre himself solved this conjecture for \(r=3\), and the authors of the current paper solved in under the assumption of the so-called Borisov-Alexeev-Borisov (BAB) conjecture. The BAB conjecture was recently solved by C. Birkar [“Singularities of linear systems and boundedness of Fano varieties”, Preprint, arXiv:1609.05543].One can ask analogs for \(\mathrm{Bir}(X)\) of more general varieties \(X\). In this direction, the authors proved that if \(X\) is a rationally connected \(3\)-fold and \(p\geq 10386\), then any finite \(p\)-subgroup of \(\mathrm{Bir}(X)\) is finite and generated by at most \(3\) elements (see Theorem 1.2). Some other results are known about optimal upper bounds for the ranks of abelian \(p\)-subgroups of \(\mathrm{Cr}_2\) or \(\mathrm{Bir}(X)\) (where \(X\) is a rationally connected \(3\)-fold) without restrictions on \(p\) (see Theorems 1.3 and 1.4).The purpose of the paper under review is to establish some refinements of the above results. The main theorem of the paper is Theorem 1.5: Let \(X\) be a rationally connected threefold and \(G\) a \(p\)-subgroup of \(\mathrm{Bir}(X)\). Suppose that \(p\geq 17\). Then \(G\) is generated by at most \(r\) elements.The same idea of the proof also allows to reprove a similar result for \(\mathrm{Bir}(X)\), where \(X\) is a rational surface.One main idea for the proof is Remark 3.4: Let \(U\) be an irreducible variety acted on by a finite group \(G\). Suppose that \(G\) has a fixed point \(p\in U\). Then there is an embedding \(G\) into \(\mathrm{GL}(T_p(U))\). This Remark can be combined with the following Lemma to reduce the cases to consider, Lemma 3.3: Let \(G\subset\mathrm{GL}_n\) be a \(p\)-group. If \(p>n\), then \(G\) is abelian and \(\mathrm{rank}(G)\leq n\).The remaining of Section 3 is to present a collection of lemmas stating the existence of fixed points for group actions if some assumptions are satisfied. In Section 4 the authors establish the existence of fixed points for liftings of groups actions under equivariant-minimal model program. Section 5 explores the situation for surfaces, and Section 6 and 7 explore the situation for (singular) Fano threefolds.The proofs of the main results are given in Section 8. By running equivariant resolution of singularities, the authors can reduce to the case \(X\) is smooth and \(G\subset\mathrm{Aut}(X)\). Then by running \(G\)-minimal model program the authors can reduce to the special cases considered in Sections 3–7. Reviewer: Tuyen Truong (Syracuse) Cited in 1 ReviewCited in 11 Documents MSC: 14E07 Birational automorphisms, Cremona group and generalizations Keywords:birational automorphisms; Cremona group; \(p\)-groups PDFBibTeX XMLCite \textit{Y. Prokhorov} and \textit{C. Shramov}, Math. Nachr. 291, No. 8--9, 1374--1389 (2018; Zbl 1423.14099) Full Text: DOI arXiv