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Register a new userMoufang loops and toric surfaces
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Authors
Vadim Schechtman
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The toric surfaces for octonions and related objects are discussed.
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We study toric varieties over an arbitrary field with an emphasis on toric surfaces in the Merkurjev-Panin motivic category of K-motives. We explore the decomposition of certain toric varieties as K-motives into products of central simple algebras, the geometric and topological information encoded in these central simple algebras, and the relationship between the decomposition of the K-motives and the semiorthogonal decomposition of the derived categories. We obtain the information mentioned above for toric surfaces by explicitly classifying all minimal smooth projective toric surfaces using toric geometry.
The Noether-Lefschetz theorem asserts that any curve in a very general surface $X$ in $mathbb P^3$ of degree $d geq 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that under some conditions, which replace the condition $d geq 4$, a very general surface in a simplicial toric threefold $mathbb P_Sigma$ (with orbifold singularities) has the same Picard number as $mathbb P_Sigma$. Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in $mathbb P_Sigma$ in a linear system of a Cartier ample divisor with respect to a (-1)-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether-Lefschetz loci which contain a line, defined as a rational curve that is minimal in a suitable sense.
Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a very general surface of degree at least five in projective space is algebraically hyperbolic. We are interested in generalizing the study of surfaces in projective space to surfaces in smooth projective toric threefolds with Picard rank 2 or 3. Following Kleinschmidt and Batyrev, we explore the combinatorial description of smooth projective toric threefolds with Picard rank 2 and 3. We then use Haase and Iltens method of finding algebraically hyperbolic surfaces in toric threefolds. As a result, we determine many algebraically hyperbolic surfaces in each of these varieties.
In 2005 Dullin et al. proved that the non-zero vector of Maslov indices is an eigenvector with eigenvalue 1 of the monodromy matrices of an integrable Hamiltonian system. We take a close look at the geometry behind this result and extend it to a more general context. We construct a bundle morphism defined on the lattice bundle of an (general) integrable system, which can be seen as a generalization of the vector of Maslov indices. The non-triviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue 1 of the monodromy matrices, and gives rise to a corank 1 toric foliation refining the original one induced by the integrable system. Furthermore, we show that in the case where the system has 2 degrees of freedom, this implies the global existence of a free S^{1} action.
We study the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety $X$ with branching set the invariant divisor under the action of $(mathbb{C}^*)^n$. This is the proalgebraic toric-completion $X_{mathbb{Q}}$ of $X$. The ramification over the invariant divisor and the singular invariant divisors of $X$ impose topological constraints on the automorphisms of $X_{mathbb{Q}}$. Considering this proalgebraic space as the toric functor on the adelic complex plane multiplicative semigroup, we calculate its automorphic group. Moreover we show that its vector bundle category is the direct limit of the respective categories of the finite toric varieties coverings defining the proalgebraic toric-completion.
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