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Filtrations, multiplicities and singularities
Graded sequences of ideals, or filtrations on local rings, play an important role in commutative algebra, the theory of singularities and birational geometry.
In this talk, I will introduce the saturation of a filtration and discuss several results about multiplicities of filtrations. The first is a theorem generalizing
Rees' theorem on Hilbert-Samuel multiplicities of ideals. Then I will talk about convexity of multiplicities along geodesics, with applications in valuation theory
and K-stability. I will also introduce a metric on the space of filtrations as an analogue of the Darvas metric in complex geometry, and prove the semiconuity of log
canonical threshold functions in the spirit of Kollár-Demailly. This talk is partially based on joint work with Harold Blum and Yuchen Liu.
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Del Pezzo surfaces of degree 4
A del Pezzo surface is a smooth projective surface with ample anticanonical divisor. Over an algebraically closed field, any surface like this is rational. However,
without this assumption del Pezzo surfaces exhibit very interesting birational properties. I will survey some old and new results about birational geometry of
del Pezzo surfaces over arbitrary fields, mostly focusing on del Pezzo surfaces of degree 4. These can be realized as intersections of two quadrics. It is known
that minimal del Pezzo surfaces of degree 4 over a perfect field are never rational. I will provide a detailed description of birational models for such surfaces,
and will discuss the properties of their birational automorphism groups. The talk is based on a joint work in progress with A.Trepalin.
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