Cremona groups and the icosahedron by Ivan Cheltsov, Constantin Shramov

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 gorgeous appearances of the icosahedral team A5 in them. The booklet surveys recognized evidence approximately surfaces with an motion of A5, explores A5-equivariant geometry of the quintic del Pezzo threefold V5, and offers an explanation 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 variety that could be a measure 5 hide of the diagonal Clebsch cubic floor. in addition they current birational selfmaps of V5 that go back and forth with A5-action and use them to figure out the full workforce of A5-birational automorphisms. because of this research, they produce 3 non-conjugate icosahedral subgroups within the Cremona team of rank three, one in all them coming up from the threefold V5.

This e-book offers updated instruments for learning birational geometry of higher-dimensional forms. specifically, it presents readers with a deep knowing of the biregular and birational geometry of V5.

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9. 1 is indeed not easy to prove. Nevertheless, we find it well motivated, since it gives us a reason both to refine the methods of proving A5 -birational rigidity, and to study a beautiful world of icosahedral symmetries of the variety V5 . The largest part of the book is occupied by such studies. 1 in Chapter 18, as well as in the closely related classification of A5 -invariant Halphen pencils on V5 in Chapter 19. However, some of the facts about A5 -equivariant geometry of V5 that we establish are not directly used in the proofs, and in general we regard such results as having their internal value and beauty.

Put P4 = P(I ⊕ W4 ). 1), and X4 be an irreducible A5 -invariant quartic in P4 . Let X4 be a double cover of Q branched 3 = 4. One can over Q ∩ X4 . Then X4 is a Fano threefold such that −KX 4 check that the action of A5 lifts to X4 . 8]. It is easy to show that a general threefold X4 is smooth. We do not know whether there are rational examples or not. 7. Put P4 = P(W5 ). 2), and X4 ⊂ P4 be an irreducible A5 -invariant quartic. 6, we can construct a double cover of Q branched over Q ∩ X4 . One can show that it is also a Fano threefold that is acted on by A5 .

Then the log pair ¯ M ¯ + a1 ∆ ¯ 1 + a2 ∆ ¯2 + S, multO M + a1 + a2 − 1 E is not Kawamata log terminal at some point Q ∈ E. Thus multO M1 · M2 ¯1 · M ¯2 . 5) To complete the proof, we must consider the following possible cases: ¯1 ∪ ∆ ¯ 2; • Q∈∆ ¯ 1; • Q=E∩∆ ¯ 2. • Q=E∩∆ ¯1 ∪ ∆ ¯ 2 . Then the log pair Suppose that Q ∈ ∆ ¯ M ¯ + S, multO M + a1 + a2 − 1 E is not Kawamata log terminal at the point Q. Thus, we may assume that ¯1 · M ¯2 multQ M 4 2 − multO M − a1 − a2 2 by induction, replacing the curve ∆1 by E, the number a1 by the number multO (M) + a1 + a2 − 1, and the number a2 by 0.

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