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Cremona Groups and the Icosahedron
CRC Press, 2015.
Cheltsov I., Shramov K.
Cremona Groups and the Icosahedron focuses on the Cremona groups of ranks 2 and 3 and describes the beautiful appearances of the icosahedral group A5 in them. The book surveys known facts about surfaces with an action of A5, explores A5-equivariant geometry of the quintic del Pezzo threefold V5, and gives a proof of its A5-birational rigidity.
The authors explicitly describe many interesting A5-invariant subvarieties of V5, including A5-orbits, low-degree curves, invariant anticanonical K3 surfaces, and a mildly singular surface of general type that is a degree five cover of the diagonal Clebsch cubic surface. They also present two birational selfmaps of V5 that commute with A5-action and use them to determine the whole group of A5-birational automorphisms. As a result of this study, they produce three non-conjugate icosahedral subgroups in the Cremona group of rank 3, one of them arising from the threefold V5.
This book presents up-to-date tools for studying birational geometry of higher-dimensional varieties. In particular, it provides readers with a deep understanding of the biregular and birational geometry of V5.
Priority areas:
mathematics
Language:
English
Publication based on the results of:
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the fundamental group of a 3-manifold M admitting an Anosov flow must
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We investigate the interplay between the dimension of the space of static potentials and the geometric and topological structure of the underlying static three-manifold. A partial classification of boundaryless static manifolds is obtained in terms of this dimension. We also treat the case of static manifolds with boundary. In particular, we prove that if a ...
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Label prediction in neural networks (NNs) has O(n) complexity proportional to the number of classes. This holds true for classification using fully connected layers and cosine similarity with some set of class prototypes. In this paper we show that if NN latent space (LS) geometry is known and possesses specific properties, label prediction complexity can ...
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We study Blaschke--Santal{ó}-type inequalities for N>=2 sets (functions) and a special class of cost functions. In particular, we prove new results about reduction of the maximization problem for the Blaschke--Santal{ó}-type functional to homogeneous case (functional inequalities on the sphere) and extend the symmetrization argument to the case of N>2 sets.
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Prokhorov Y., Pinardin A., Cheltsov I., / Series arXiv "math". 2025.
We show that the alternating groups $\mathfrak{A}_5$ and $\mathfrak{A}_6$ are the only finite simple non-abelian subgroups of the group of birational selfmaps of the real three-dimensional projective space. ...
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Loginov K., Zhang Z., Pinardin A., / Series arXiv "math". 2025.
We study finite abelian groups acting on three-dimensional rationally connected varieties. We concentrate on the groups of K3 type, that is, abelian extensions by a cyclic group of groups that faithfully act on a K3 surface. In particular, if a finite abelian group faithfully acts on a threefold preserving a K3 surface (with at worst ...
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Arzhantsev I., Quaestiones Mathematicae 2024 Vol. 47 No. 9 P. 1767 –1774
An additive action on an irreducible algebraic variety X is an effective action with an open orbit of the vector group . Any two additive actions on X are conjugate by a birational automorphism of X. We prove that, if X is the projective space, the conjugating element can be chosen in the affine Cremona group and it is given by so-called basic polynomials ...
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Arzhantsev I., Zaidenberg M., Proceedings of the American Mathematical Society 2023 Vol. 151 No. 7 P. 2813–2829
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Avilov A., / Series math "arxiv.org". 2022.
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Kolotova L., Norman G., Pisarev V., Journal of Non-Crystalline Solids 2015 Vol. 429 P. 98–103
Molecular dynamics study of transition from liquid aluminum into an amorphous state is carried out. Different
criteria for the glass transition are compared with each other: splitting of the second peak of the radial distribution
function, increasing of the number of atoms with icosahedron-like environment, increase of the selfdiffusion
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