Abstract
The aim of these notes is to describe some relations between the birational geometry of algebraic varieties and their associated derived categories. This is a large subject with several diverging paths, so we’ll restrict our focus to the realm of topics discussed at the Alberta superschool. Being lecture notes, the discussion here is somewhat informal, and we only attempt a general overview, and referring the reader to the relevant original research articles for details. Given the background of the participants of the Alberta superschool, these notes take the somewhat unorthodox approach of assuming that the reader has modest familiarity with derived and triangulated categories, but is perhaps not as familiar with the more cabalistic aspects of birational geometry.
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Notes
- 1.
Here “mild” actually means normal and \(\mathbb {Q}\)-factorial. The author apologizes for occasionally taking the liberty to be relaxed about types of singularities in this article, despite their fundamental role in birational geometry.
- 2.
See e.g. Theorem 6.15 in [12].
- 3.
I.e. \(-K_X\) is ample.
- 4.
I won’t define “extremal” except to say that it’s a suitable generalization of being a \({-}1\) curve on a surface. The definition is not hard, but formulating it is a bit orthogonal to our purposes in these notes.
- 5.
I.e. is Y is not \(\mathbb {Q}\)-factorial.
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Diemer, C. (2018). Birational Geometry and Derived Categories. In: Ballard, M., Doran, C., Favero, D., Sharpe, E. (eds) Superschool on Derived Categories and D-branes. SDCD 2016. Springer Proceedings in Mathematics & Statistics, vol 240. Springer, Cham. https://doi.org/10.1007/978-3-319-91626-2_7
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