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Title: Derived categories of complete intersections, crepant categorical resolutions and homological projective duality
Author: Turcinovic, Zak
ISNI:       0000 0004 7658 8584
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2019
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Homological projective duality describes the bounded derived category D(X) of coherent sheaves on X when the latter is a complete intersection of hypersurfaces of the same degree. We give a procedure for constructing new HP duals from old and move beyond HPD by studying complete intersections cut out by hypersurfaces of varying degrees. First, starting with a HP dual pair X, Y and smooth orthogonal linear sections X_{L^{\perp}} , Y_L, we prove that the blowup of X in X_{L^{\perp}} is naturally HP dual to Y_L. The result also holds true when Y is a noncommutative variety or just a category. We extend the result to the case where the base locus X_{L^{\perp}} is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a weakly crepant categorical resolution of singularities of Y_L. Second, for a complete intersection X ⊂ P^n, we show that the nontrivial part of D(X) is a weakly crepant categorical resolutions of a certain fractional Calabi-Yau category. This turns out to give examples of the previous story where, starting with a noncommutative HP dual, the blowing up process nevertheless gives geometric HP duals. This is also a generalisation of Kuznetsov's story for nodal cubic fourfolds [32] and in particular we obtain new examples of Calabi-Yau categories that admit a geometric crepant resolution. Furthermore, we get a characterisation of complete intersections of Calabi-Yau Hodge type in terms of their derived category and infinitely many examples of rational varieties that are categorically representable in codimension 2.
Supervisor: Thomas, Richard Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral