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Algebraic Legendrian varieties

100%
Buczyński J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity (see [Arn74], [Sła91] and references therein). However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space; prior to the author's research only few smooth examples of these were known (see [Bry82], [LM07]). Strong restrictions on the topology of such varieties have been found and studied by Landsberg and Manivel ([LM07]). This dissertation reviews the subject of Legendrian varieties and extends some of recent results. The first series of results is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable minor assumptions) is completely determined by the sections of the distinguished line bundle on the contact manifold vanishing on the Legendrian variety. Moreover, its action preserves the contact structure. The relation between the Lie algebra tangent to automorphisms and the sections is given by an explicit formula (see also [LeB95], [Bea07]). This summarises and extends some earlier results of the author. The second series of results is devoted to finding new examples of smooth Legendrian subvarieties of projective space. The examples found by other researchers were some homogeneous spaces, many examples of curves and a family of surfaces birational to some K3 surfaces. Further the author found a couple of other examples including a smooth toric surface and a smooth quasihomogeneous Fano 8-fold. Finally, the author proved that both of these are special cases of a very general construction: a general hyperplane section of a smooth Legendrian variety, after a suitable projection, is a smooth Legendrian variety of smaller dimension. We review all of those examples and also add infinitely many new examples in every dimension, with various Picard rank, canonical degree, Kodaira dimension and other invariants. The original motivation for studying complex Legendrian varieties comes from a 50 years old problem of giving compact examples of quaternion-Kähler manifolds (see [Ber55], [LS94], [LeB95] and references therein). Also Legendrian varieties are related to some algebraic structures (see [Muk98], [LM01], [LM02]). A new potential application to classification of smooth varieties with smooth dual arises in this dissertation.
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Maps of toric varieties in Cox coordinates

61%
Brown G., Buczyński J.
Fundamenta Mathematicae
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2013
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tom 222
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nr 3
213-267
EN
The Cox ring provides a coordinate system on a toric variety analogous to the homogeneous coordinate ring of projective space. Rational maps between projective spaces are described using polynomials in the coordinate ring, and we generalise this to toric varieties, providing a unified description of arbitrary rational maps between toric varieties in terms of their Cox coordinates. Introducing formal roots of polynomials is necessary even in the simplest examples.
3
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On the graph labellings arising from phylogenetics

49%
Buczyńska W., Buczyński J., Kubjas K., Michałek M.
Open Mathematics
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2013
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tom 11
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nr 9
1577-1592
EN
We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.
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