ATAG_Algebraic Torus Actions: Geometry and Combinatorics
Final Report Abstract
The present project concerns fundamental research in the field of algebraic geometry, a central branch of modern mathematics which has many vital applications in other branches of mathematics, theoretical physics, computational biology, computer science and engineering. The theory of algebraic varieties with algebraic torus action is a vast and active research field on the border of algebraic geometry, topology, representation theory and discrete mathematics. This project has extended the applicability of methods established in relation to equivariant cohomology and toric geometry in order to understand the geometry of algebraic varieties with torus action. The main novel findings concern deformation theory, sheaves and cohomology as well as birational geometry. In each case, there are either combinatorial techniques developed and employed, or combinatorial problems are solved using tools from algebraic geometry.
Publications
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Polyhedral adjunction theory. Algebra & Number Theory, 7(10), 2417-2446.
Di Rocco, Sandra; Haase, Christian; Nill, Benjamin & Paffenholz, Andreas
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The dualizing sheaf on first-order deformations of toric surface singularities. Journal für die reine und angewandte Mathematik (Crelles Journal), 2019(753), 137-158.
Altmann, Klaus & Kollár, János
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A note on discrete mixed volume and Hodge–Deligne numbers. Advances in Applied Mathematics, 104, 1-13.
Di Rocco, Sandra; Haase, Christian & Nill, Benjamin
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Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP. Annals of Combinatorics, 23(2), 255-262.
Beck, Matthias; Haase, Christian; Higashitani, Akihiro; Hofscheier, Johannes; Jochemko, Katharina; Katthän, Lukas & Michałek, Mateusz
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Stringy -functions of canonical toric Fano threefolds and their applications. Izvestiya: Mathematics, 83(4), 676-697.
Batyrev, V. V. & Schaller, K.
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Displaying the cohomology of toric line bundles. Izvestiya: Mathematics, 84(4), 683-693.
Altmann, K. & Ploog, D.
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Fujita’s Freeness Conjecture for T-Varieties of Complexity One. Michigan Mathematical Journal, 69(2).
Altmann, Klaus & Ilten, Nathan
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Toric Newton-Okounkov functions with an application to the rationality of certain Seshadri constants on surfaces. 2020
Christian Haase, Alex Küronya & Lena Walter
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Algebraic hyperbolicity for surfaces in toric threefolds. Journal of Algebraic Geometry, 30(3), 573-602.
Haase, Christian & Ilten, Nathan
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Existence of unimodular triangulations — positive results. Memoirs of the American Mathematical Society, 270(1321).
Haase, Christian; Paffenholz, Andreas; Piechnik, Lindsey & Santos, Francisco
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Mirror symmetry for quasi-smooth Calabi–Yau hypersurfaces in weighted projective spaces. Journal of Geometry and Physics, 164, 104198.
Batyrev, Victor & Schaller, Karin
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The structure of exceptional sequences on toric varieties of Picard rank two
Klaus Altmann & Frederik Witt
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Toric co-Higgs sheaves. Journal of Pure and Applied Algebra, 225(8), 106634.
Altmann, Klaus & Witt, Frederik
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Exceptional sequences of 8 line bundles on (P^1)^3. Journal of Algebraic Combinatorics, 56(2), 305-322.
Altmann, Klaus & Altmann, Martin
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NObodies are perfect, their semigroups are not
Klaus Altmann
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On the Fine Interior of Three-Dimensional Canonical Fano Polytopes. Springer Proceedings in Mathematics & Statistics, 11-47. Springer International Publishing.
Batyrev Victor; Kasprzyk Alexander & Schaller Karin
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Polyhedra, lattice structures, and extensions of semigroups. Journal of the London Mathematical Society, 106(4), 3938-4008.
Altmann, Klaus; Constantinescu, Alexandru & Filip, Matej
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Versality in toric geometry. Journal of Algebra, 609, 1-43.
Altmann, Klaus; Constantinescu, Alexandru & Filip, Matej
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Extensions of toric line bundles. Mathematische Zeitschrift, 304(1).
Altmann, Klaus; Flatt, Amelie & Hille, Lutz
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LDP polygons and the number 12 revisited
Ulrike Böcking
