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Volume 214, Nº 6 (2023)

Ano: 2023
Artigos: 6
URL: https://journal-vniispk.ru/0368-8666/issue/view/7503

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Somente assinantes

Birational rigidity of G-del Pezzo threefolds of degree 2

Avilov A.

Resumo

We classify nodal rational non- $Q$ -factorial del Pezzo threefolds of degree $2$ which can be $G$ -birationally rigid for some subgroup $G \subset Aut (X)$

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Matematicheskii Sbornik. 2023;214(6):3-40

3-40

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How is a graph not like a manifold?

Ayzenberg A., Masuda M., Solomadin G.

Resumo

For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S (X)$ of face submanifolds. We prove that the condition of the $j$ -independency of tangent weights at each fixed point implies the $(j + 1)$ -acyclicity of the skeleta $S (X)_{r}$ for $r > j + 1$ . This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2 n$ with an $(n - 1)$ -independent action of the $(n - 1)$ -dimensional torus, under certain colourability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. This observation underlines a certain similarity between actions of complexity $1$ and torus manifolds.

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Matematicheskii Sbornik. 2023;214(6):41-68

41-68

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Forms of del Pezzo surfaces of degree 5 and 6

Zaitsev A.

Resumo

We obtain necessary and sufficient condition for the existence of del Pezzo surfaces of degrees $5$ and $6$ over a field $K$ with a prescribed action of absolute Galois group $Gal (K^{s e p} / K)$ on the graph of $(- 1)$ -curves. We also compute the automorphism groups of del Pezzo surfaces of degree $5$ over arbitrary fields.

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Matematicheskii Sbornik. 2023;214(6):69-86

69-86

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Algebra of shares, complete bipartite graphs, and $\mathfrak{sl}_2$ weight system

Zinova P., Kazarian M.

Resumo

A function of chord diagrams is called a weight system if it satisfies the so-called four-term relations. Vassiliev's theory describes finite-order knot invariants in terms of weight systems. In particular, there is a weight system corresponding to the coloured Jones polynomial. This weight system is described in terms of the Lie algebra ${s l}_{2}$ . According to the Chmutov-Lando theorem, the value of this weight system depends only on the intersection graph of the chord diagram. Therefore, it is possible to discuss the values of this weight system at intersection graphs.
We obtain formulae for the generating functions of the values of the ${s l}_{2}$ weight system at complete bipartite graphs. Using these formulae we prove that Lando's conjecture about the degree of the polynomial that is the value of this weight system at the projection of a graph onto the subspace of primitive elements in the Hopf algebra of graphs is true for complete bipartite graphs and for a certain wider class of graphs.
We introduce the algebra of shares and the ${s l}_{2}$ weight system on shares. These are the main tools for our proof.

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Matematicheskii Sbornik. 2023;214(6):87-109

87-109

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Comparison theorems for evolution inclusions with maximal monotone operators. $L^2$-theory

Tolstonogov A.

Resumo

An evolution inclusion with time-dependent family of maximal monotone operators in considered in a separable Hilbert space. If the elements with minimum norm of the family of maximal monotone operators satisfy certain growth conditions, then the domains of definition of this family are closed convex sets. Hence the sweeping process is well defined, whose values are the normal cones of the domains of definition of maximal monotone operators. It is shown that if the sweeping process has a solution for each single-valued perturbation from the space of integrable functions, then the evolution inclusion with the maximal monotone operators and single-valued perturbations from the space of integrable functions is also solvable. Quite general conditions in terms of the properties of the family of maximal monotone operators that ensure the existence of solutions for the sweeping process are presented.
All results obtained and the approach presented are new. They are used to prove an existence theorem for evolution inclusions with multivalued perturbations, whose values are closed nonconvex sets.

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Matematicheskii Sbornik. 2023;214(6):110-135

110-135

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On the multiplicative Chung–Diaconis–Graham process

Shkredov I.

Resumo

We study the lazy Markov chain on $F_{p}$ defined as follows: $X_{n + 1} = X_{n}$ with probability $1 / 2$ and $X_{n + 1} = f (X_{n}) \cdot ε_{n + 1}$ , where the $ε_{n}$ are random variables distributed uniformly on the set ${γ, γ^{- 1}}$ , $γ$ is a primitive root and $f (x) = x / (x - 1)$ or $f (x) = i n d (x)$ . Then we show that the mixing time of $X_{n}$ is $\exp (O (\log p \cdot \log \log \log p / \log \log p))$ . Also, we obtain an application to an additive-combinatorial question concerning a certain Sidon-type family of sets.

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Matematicheskii Sbornik. 2023;214(6):136-154

136-154

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