| Abstract: |
Rational curves and their neighbourhoods in surfaces and 3-folds provide a playground for rich interplay of algebra and geometry, the first instance of such interaction being Beilinson’s observation that the derived category of a projective line admits infinitely many algebraic t-structures. We can walk between these, one step at a time, using various techniques of tilting, mutation, and application of the Picard group action. Then upon iterating these operations, a `fixed—point theorem’ reveals itself: the geometric category of coherent sheaves is naturally a limit of algebraic hearts. I will describe the convex—geometric and combinatorial tools used to study the result, and how it is used to classify t-structures and spherical objects with small cohomological spread in the derived category of a flopping curve in a 3-fold.
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