Linear systems on fibrations

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This award supports a research project in the field of algebra and geometry, old and traditional areas of mathematics, with methods and applications in modern birational geometry. These methods and applications interact with most of branches of mathematics, including differential geometry, topology, number theory, and algebra, and can be useful in mathematical physics, cosmology, cryptography, and robotics. From its origin geometry is closely related to volume. The main objects under investigation are differential forms of volume type on families and their products. The project addresses problems on polynomial finite generatedness of those forms and on estimation of degrees of generators with respect to the dimension and possibly to some other discrete invariants of members of families. New techniques are needed in dealing with differential forms on families or relative differential forms. Another novelty of the project is to develop theory of the finite generatedness based on universal mappings from moduli theory instead of traditionally used vanishing of cohomologies.The project deals with fundamental problems on linear systems of divisors associated to twisted differential forms on fibered spaces. Standard problems about linear systems are problems about their base points, about singularities of their general or special members, and about correspondences with or isomorphisms to other systems. Respectively, the PI considers the semiampleness problem for moduli part of weakly log canonical families, the construction of log canonical complements for fibered varieties with applications to log canonical local and global thresholds, to a new beta invariant and to Tian's alpha invariant, and explores possibility of an isomorphism between log canonical linear systems adjoint to the discriminant curve for different conic bundles structures of a threefold with applications to birational classification of those conic bundles. Strictly fibered cases for log canonical complements and for Sarkisov's links do not cover all possible cases, they cover main cases and remaining ones are exceptional. The latter form usually bounded families up to birational isomorphisms. A new technique relevant to these problems will be developed under the project. This makes use of relative toroidal log singularities, theory of b-polarizations, theory of moduli of triples, interlaced by flops, and correspondences of those moduli. Investigation of moduli of varieties defined up to flops is natural and agrees with the spirit of current birational geometry where the minimal models are defined up to flops.