Онлайн лекции из серии научных презентаций исследований молодых ученых "Новые имена"

For a blow-up of a smooth point on an algebraic surface we can give a rather simple description of its derived category of coherent sheaves: take the derived category of the original surface, take a single object corresponding to the exceptional divisor, and glue them together in a natural way. This parallels the relation between Picard groups, and forms the basis of the analogy between semi-orthogonal decompositions and the minimal model program. This analogy is not an exact correspondence, but in lower dimensions there are specific results and conjectures. I will discuss some general observations and explain why a smooth projective surface with a nef and effective canonical class has indecomposable derived category, confirming the conjecture of Okawa.

In this talk I will present a relative monoidal version of the Bondal-Orlov reconstruction theorem. Concretely the goal is to put in contrast this classical result with Balmers reconstruction theorem. Using techniques of higher category theory we construct a sheaf of bifunctors over a smooth projective variety X faithfully flat over a quasicompact quasiseparated scheme S. When every fiber over S has an ample canonical bundle we observe that a tensor structure on the derived category of X must under some reasonable conditions be equivalent to the one given by the usual derived tensor product. This is joint work with Artan Sheshmani