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In this talk, I will give a brief survey on singularities and log pairs in birational geometry. Log terminal singularities, log canonical singularities, rational singularities, and Du Bois singularities naturally appear in many areas in algebraic geometry such as birational geometry, moduli theory, and Hodge theory.
Grothendieck posed a question of whether the natural map from the Brauer group of a scheme to its cohomological one is an isomorphism of abelian groups. It’s not true in general, but we have some positive results from Grothendieck and Gabber (and de Jong), among many others. After a brief review of Brauer groups in algebraic geometry, I’ll talk about some recent progress in the setting of derived and spectral algebraic geometry, where we can provide an affirmative answer for quasi-compact and quasi-separated (derived/spectral) schemes, and my work which extends the previous results to spectral algebraic stacks.
In this talk, I will show that secant varieties of a smooth projective curve embedded by a sufficiently large degree has normal Cohen-Macaulay Du Bois singularities. I will also prove that the curve is rational if and only if any secant variety has log terminal singularities, and the curve is elliptic if and only if any secant variety has log canonical singularities that are not log terminal (not even rational). This talk is based on joint work with Lawrence Ein and Wenbo Niu.
Ulrich complexity for a given projective variety X, originally introduced to measure the complexity of polynomials by Bläser-Eisenbud-Schreyer, is defined as the smallest possible rank for the Ulrich sheaves on X. The existence of an Ulrich sheaf on any hypersurface is well-known, however, Ulrich complexity is not very well understood even for cubic hypersurfaces. In this talk, I will review some recent studies on Ulrich complexity for small cubics, in particular, for smooth cubic fourfolds. This is a joint work in progress with D. Faenzi.
Derived/spectral algebraic geometry is a relatively new area which features homotopy theory in algebraic geometry. I’ll take deformation theory and intersection theory to provide some flavor of these new fields. There are no prerequisites required other than ordinary algebraic geometry, so everyone is welcome to attend. (There are two lectures; I, II. This is the second of them.)