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Bull. Korean Math. Soc. 2017; 54(5): 0~1857
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2017-09-30
Okounkov bodies and Zariski decompositions on surfaces
Sung Rak Choi, Jinhyung Park, Joonyeong WonAbstract : The purpose of this paper is to investigate the close relation between Okounkov bodies and Zariski decompositions of pseudoeffective divisors on smooth projective surfaces. Firstly, we completely determine the limiting Okounkov bodies on such surfaces, and give applications to Nakayama constants and Seshadri constants. Secondly, we study how the shapes of Okounkov bodies change as we vary the divisors in the big cone.
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2017-09-30
Second Chern numbers of vector bundles and higher adeles
Denis V. OsipovAbstract : We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic $K$-theory and depends on the canonical $\Z$-torsor of a locally linearly compact $k$-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed.
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2017-09-30
Efficiently computing torus charts in Landau--Ginzburg models of complete intersections in Grassmannians of planes
Thomas PrinceAbstract : In this note, companion to the paper~\cite{PSh14b}, we describe an alternative method for finding Laurent polynomials mirror-dual to complete intersections in Grassmannians of planes, in the sense discussed in \cite{PSh14b}. This calculation follows a general method for finding torus charts on Hori--Vafa mirrors to complete intersections in toric varieties, detailed in \cite{CKP14} generalising the method of \cite{Prz10}.
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2017-09-30
The Bogomolov-Prokhorov invariant of surfaces as equivariant cohomology
Evgeny ShinderAbstract : For a complex smooth projective surface $M$ with an action of a finite cyclic group $G$ we give a uniform proof of the isomorphism between the invariant $H^1(G, H^2(M, \Z))$ and the first cohomology of the divisors fixed by the action, using $G$-equivariant cohomology. This generalizes the main result of Bogomolov and Prokhorov \cite{BP}.
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2017-09-30
Endomorphisms of projective bundles over a certain class of varieties
Ekaterina Amerik, Alexandra KuznetsovaAbstract : Let $B$ be a simply-connected projective variety such that the first cohomology groups of all line bundles on $B$ are zero. Let $E$ be a vector bundle over $B$ and $X=\p(E)$. It is easily seen that a power of any endomorphism of $X$ takes fibers to fibers. We prove that if $X$ admits an endomorphism which is of degree greater than one on the fibers, then $E$ splits into a direct sum of line bundles.
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2017-09-30
Hilbert's Theorem 90 for non-compact groups
Marat RovinskyAbstract : Let $K$ be a field and $G$ be a group of its automorphisms. It follows from Speiser's generalization of Hilbert's Theorem 90, \cite{Speiser} that any $K$-{\it semilinear} representation of the group $G$ is isomorphic to a direct sum of copies of $K$, if $G$ is finite. In this note three examples of pairs $(K,G)$ are presented such that certain irreducible $K$-semilinear representations of $G$ admit a simple description: (i) with precompact $G$, (ii) $K$ is a field of rational functions and $G$ permutes the variables, (iii) $K$ is a universal domain over field of characteristic zero and $G$ its automorphism group. The example (iii) is new and it generalizes the principal result of \cite{adm}.
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2017-09-30
Singularities of divisors on flag varieties via Hwang's product theorem
Evgeny SmirnovAbstract : We give an alternative proof of a recent result by B.~Pasquier stating that for a generalized flag variety $X=G/P$ and an effective $\QQ$-divisor $D$ stable with respect to a Borel subgroup the pair~\mbox{$(X,D)$} is Kawamata log terminal if and only if~\mbox{$\lfloor D\rfloor=0$}.
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2017-09-30
Minimal del Pezzo surfaces of degree 2 over finite fields
Andrey TrepalinAbstract : Let $X$ be a minimal del Pezzo surface of degree $2$ over a finite field $\F_q$. The image $\Gamma$ of the Galois group $\Gal(\overline{\F}_q / \F_q)$ in the group $\Aut(\Pic(\XX))$ is a cyclic subgroup of the Weyl group $W(E_7)$. There are $60$ conjugacy classes of cyclic subgroups in $W(E_7)$ and $18$ of them correspond to minimal del Pezzo surfaces. In this paper we study which possibilities of these subgroups for minimal del Pezzo surfaces of degree $2$ can be achieved for given $q$.
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2017-09-30
Equivariant matrix factorizations and Hamiltonian reduction
Sergey Arkhipov, Tina KanstrupAbstract : Let $X$ be a smooth scheme with an action of an algebraic group $G$. We establish an equivalence of two categories related to the corresponding moment map $\mu : T^*X \to \g^*$ - the derived category of $G$-equivariant coherent sheaves on the derived fiber $\mu^{-1}(0)$ and the derived category of $G$-equivariant matrix factorizations on $T^*X \times \g$ with potential given by $\mu$.
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2017-09-30
Integral Chow motives of threefolds with $K$-motives of unit type
Sergey GorchinskiyAbstract : We prove that if a smooth projective algebraic variety of dimension less or equal to three has a unit type integral $K$-motive, then its integral Chow motive is of Lefschetz type. As a consequence, the integral Chow motive is of Lefschetz type for a smooth projective variety of dimension less or equal to three that admits a full exceptional collection.
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