By Ivan Cheltsov, Constantin Shramov
Cremona teams and the Icosahedron makes a speciality of the Cremona teams of ranks 2 and three and describes the attractive appearances of the icosahedral team A5 in them. The ebook surveys identified proof approximately surfaces with an motion of A5, explores A5-equivariant geometry of the quintic del Pezzo threefold V5, and offers an evidence of its A5-birational rigidity.
The authors explicitly describe many fascinating A5-invariant subvarieties of V5, together with A5-orbits, low-degree curves, invariant anticanonical K3 surfaces, and a mildly singular floor of common style that may be a measure 5 conceal of the diagonal Clebsch cubic floor. additionally they current birational selfmaps of V5 that shuttle with A5-action and use them to figure out the total team of A5-birational automorphisms. due to this examine, they produce 3 non-conjugate icosahedral subgroups within the Cremona workforce of rank three, one in every of them bobbing up from the threefold V5.
This booklet offers updated instruments for learning birational geometry of higher-dimensional types. particularly, it presents readers with a deep knowing of the biregular and birational geometry of V5.
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Additional resources for Cremona groups and the icosahedron
After this it is often possible to exclude most possibilities by considering some explicit intersections. In Chapter 17 we prove auxiliary facts about base loci of various A5 -invariant linear systems that will be used later in the process of isolation of centers of bad singularities. 1 we show that the points of A5 -orbits of length 10 in V5 are in some sense in sufficiently general position. 2 we use this information to determine the base locus of the unique two-dimensional A5 -invariant subsystem of the anticanonical linear system on V5 , which helps to isolate several A5 -orbits of small length.
In particular, it appears that many such curves are contained in the surfaces of the pencil Q2 . This allows us to demonstrate that there is only a finite number of A5 -invariant curves of degree less than 30 and justifies our further attempts to obtain a classification of invariant curves of low degree. In Chapter 9 we describe combinatorial properties of A5 -invariant curves in V5 that are unions of small numbers of lines or conics. In particular, we show that some of them are disjoint unions of lines or conics, and describe intersections in several other situations.
Moreover, for every ample Q-Cartier Q-divisor ∆ on X, there exists an effective Q-divisor BZ on the variety Z such that KX + BX + ∆ Z ∼Q KZ + BZ , and the log pair (Z, BZ ) has Kawamata log terminal singularities. Suppose, in addition, that X is faithfully acted on by a finite group G such that the boundary BX is G-invariant. Note that this does not mean that each Bi is G-invariant or that each Mj is G-invariant, since the group G can permute the divisors B1 , . . , Br and it can permute the linear systems M1 , .
Cremona groups and the icosahedron by Ivan Cheltsov, Constantin Shramov