Àííîòàöèÿ:
There is an old idea that birational geometry (in particular, the Minimal Model Program)
can and should be translated into the language of derived categories. We revive this idea
by constructing “atomic” semi-orthogonal decompositions of derived categories of smooth
algebraic surfaces, over any perfect field or in $G$-equivariant setting. “Atoms” of these
decompositions provide birational invariants of surfaces, similar to invariants studied by
Lin–Shinder–Zimmermann and Auel–Bernardara. After constructing “atomic” theory, I will
explain the role that “atoms” can play in the classification of geometrically rational surfaces
over non-closed fields. This is a joint work with Evgeny Shinder and Julia Schneider, and
the idea goes back to Kontsevich and quantum cohomology.
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