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Volume 83, Nº 3 (2019)
- Ano: 2019
- Artigos: 9
- URL: https://bakhtiniada.ru/1607-0046/issue/view/7538
Articles
Vasilii Alekseevich Iskovskikh
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(3):3-4
3-4
Biregular and birational geometry of quartic double solids with 15 nodes
Resumo
Three-dimensional del Pezzo varieties of degree $2$ are double covers of$\mathbb{P}^{3}$ branched in a quartic. We prove that if a del Pezzo varietyof degree $2$ has exactly $15$ nodes, then the corresponding quartic is a hyperplanesection of the Igusa quartic or, equivalently, all such del Pezzovarieties are members of a particular linear system on the Coble fourfold.Their automorphism groups are induced from the automorphism group of theCoble fourfold. We also classify all birationally rigid varieties of thistype.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(3):5-14
5-14
Automorphisms of cubic surfaces in positive characteristic
Resumo
We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(3):15-92
15-92
Division by 2 on odd-degree hyperelliptic curves and their Jacobians
Resumo
Let $K$ be an algebraically closed field of characteristic differentfrom $2$, $g$ a positive integer, $f(x)$ a polynomial of degree $2g+1$with coefficients in $K$ and without multiple roots,$\mathcal{C}\colon y^2=f(x)$ the corresponding hyperelliptic curve ofgenus $g$ over $K$, and $J$ its Jacobian. We identify $\mathcal{C}$ withthe image of its canonical embedding in $J$ (the infinite point of$\mathcal{C}$ goes to the identity element of $J$). It is well known thatfor every $\mathfrak{b} \in J(K)$ there are exactly $2^{2g}$ elements$\mathfrak{a}\in J(K)$ such that $2\mathfrak{a}=\mathfrak{b}$. Stollconstructed an algorithm that provides the Mumford representationsof all such $\mathfrak{a}$ in terms of the Mumford representation of$\mathfrak{b}$. The aim of this paper is to give explicit formulaefor the Mumford representations of all such $\mathfrak{a}$ in terms ofthe coordinates $a,b$, where $\mathfrak{b}\in J(K)$ is given by a point$P=(a,b) \in \mathcal{C}(K)\subset J(K)$. We also prove that if $g>1$,then $\mathcal{C}(K)$ does not contain torsion points of ordersbetween $3$ and $2g$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(3):93-112
93-112
Stably rational surfaces over a quasi-finite field
Resumo
Let $k$ be a field and $X$ a smooth, projective,stably $k$-rational surface. If $X$ is split by a cyclic extension(for example, if the field $k$ is finite or, more generally, quasi-finite),then the surface $X$ is $k$-rational.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(3):113-126
113-126
Embedding derived categories of Enriques surfaces in derived categories of Fano varieties
Resumo
We show that the bounded derived category of coherent sheaves on a generalEnriques surface can be realized as a semi-orthogonal component in thederived category of a smooth Fano variety with diagonal Hodge diamond.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(3):127-132
127-132
Asymptotic bounds for spherical codes
Resumo
The set of all error-correcting codes $C$ over a fixed finite alphabet$\mathbf{F}$ of cardinality $q$ determines the set of code points in the unit square $[0,1]^2$ with coordinates $(R(C), \delta (C))$:= (relative transmission rate, relative minimal distance). The central problemof the theory of such codes consists in maximising simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in inventing explicit constructions of “good codes” and comparing new classes of codes with earlier ones.A less classical approach studies the geometry of the whole set of code points $(R,\delta)$ (with $q$ fixed), at first independently of its computability properties, and only afterwards turningto problems of computability, analogies with statistical physics, and so on.The main purpose of this article consists in extending this latter strategy to the domain of spherical codes.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(3):133-157
133-157
Threefold extremal curve germs with one non-Gorenstein point
Resumo
An extremal curve germ is the analytic germ of a threefold with terminalsingularities along a reduced complete curve admitting a contraction whosefibres have dimension at most one. The aim of the present paper is to reviewthe results concerning contractions whose central fibre is irreducible andcontains only one non-Gorenstein point.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(3):158-212
158-212
On the standard conjecture for a fibre product of three elliptic surfaces with pairwise-disjoint discriminant loci
Resumo
We prove that the Grothendieck standard conjecture $B(X)$ of Lefschetz typeon the algebraicity of the operator $ ^{\mathrm{c}}\Lambda$ of Hodge theoryis true for the fibre product $X=X_1\times_CX_2\times_CX_3$ of complex ellipticsurfaces $X_k\to C$ over a smooth projective curve $C$ provided that thediscriminant loci $\{\delta\in C\mid \operatorname{Sing}(X_{k\delta})\neq\varnothing\}$ $(k=1,2,3)$ are pairwise disjoint.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(3):213-256
213-256