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 staff A5 in them. The publication surveys identified evidence 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 attention-grabbing A5-invariant subvarieties of V5, together with A5-orbits, low-degree curves, invariant anticanonical K3 surfaces, and a mildly singular floor of normal variety that may be a measure 5 disguise 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 complete team 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 booklet offers up to date instruments for learning birational geometry of higher-dimensional forms. specifically, it offers readers with a deep realizing of the biregular and birational geometry of V5.

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Suppose that both ∆1 and ∆2 are smooth at O, and ∆1 intersects the curve ∆2 transversally at the point O. Finally, suppose that S, M + a1 ∆1 + a2 ∆2 is not Kawamata log terminal at the point O. Then  4 1 − a1 1 − a2    if a1 0 or a2 0, 2 multO M1 · M2    4 1 − a1 − a2 if a1 0 and a2 0. 4) is an equality, then multO M = 2(a1 − 1) , the log pair (S, M + a1 ∆1 + a2 ∆2 ) is log canonical, and a1 = a2 0. Proof. Let φ : S¯ → S be a blow-up of the point O, let E be the exceptional curve of φ. Then ¯ + a1 ∆ ¯ 1 + a2 ∆ ¯2 + KS¯ + M multO M + a1 + a2 − 1 E ∼Q ∼Q φ∗ KS + M + a1 ∆1 + a2 ∆2 , ¯ ∆ ¯ 1, ∆ ¯ 2 are proper transforms of M, ∆1 , ∆2 on the surface S, ¯ where M, respectively.

3 we recall the definitions and main properties of multiplier ideal sheaves, that provide an important tool to work with log canonical singularities. In particular, we discuss Nadel’s vanishing theorem and related connectedness result. 4 we discuss further properties of centers of log canonical singularities, mostly based on Kawamata’s subadjunction theorem. 5 we formulate a local inequality due to Alessio Corti concerning the multiplicities of linear systems on threefolds, and give its proof. Chapter 3 deals with birational transformations between G-Fano varieties and G-Mori fiber spaces for an arbitrary finite group G, and some varieties with similar properties.

2 we formulate and prove a slightly generalized version of Noether–Fano inequalities for GFano varieties, that is the main tool to work with birational rigidity. In our setting we start with a weak G-Fano variety, and describe a property of linear systems on it that gives an obstruction for existence of G-birational maps to other weak G-Fano varieties and fibrations with relatively ample or relatively trivial canonical class. As with the classical Noether–Fano inequalities, this property is, roughly speaking, a requirement that mobile linear systems of small degree cannot have large singularities.

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