6,789 research outputs found
Fourier–Mukai functors: a survey
No full textThis paper surveys some recent results about Fourier--Mukai functors. In particular, given an exact functor between the bounded derived categories of coherent sheaves on two smooth projective varieties, we deal with the question whether this functor is of Fourier--Mukai type. Several related questions are answered and many open problems are stated
The Beilinson complex and canonical rings of irregular surfaces
No full textAn important theorem by Beilinson, describing the bounded derived category of coherent sheaves on P^n, is extended to every weighted projective space P(w). To this purpose we consider, instead of the usual category of coherent sheaves on P(w), a suitable category of graded coherent sheaves. The weighted version of Beilinson's theorem is then applied to prove a structure theorem for good birational weighted canonical projections of surfaces of general type. This is a generalization of a theorem by Catanese and Schreyer (who treated the case of projections into P^3), and is mainly interesting for irregular surfaces, since in the regular case a similar but simpler result (due to Catanese) was already known. The result is then used to study a family of surfaces with numerical invariants p_g=q=2, K^2=4, projected into P(1,1,2,3)
A tour about existence and uniqueness of dg enhancements and lifts
No full textThis paper surveys the recent advances concerning the relations between triangulated (or derived) categories and their dg enhancements. We explain when some interesting triangulated categories arising in algebraic geometry have a unique dg enhancement. This is the case, for example, for the unbounded derived category of quasi-coherent sheaves on an algebraic stack or for its full triangulated subcategory of perfect complexes. Moreover we give an account of the recent results about the possibility to lift exact functors between the bounded derived categories of coherent sheaves on smooth schemes to dg (quasi-)functors
NON-UNIQUENESS OF FOURIER–MUKAI KERNELS
No full textWe prove that the kernels of Fourier-Mukai functors are not unique in general. On the other hand we show that the cohomology sheaves of those kernels are unique. We also discuss several properties of the functor sending an object in the derived category of the product of two smooth projective schemes to the corresponding Fourier-Mukai functor
Derived autoequivalences and a weighted Beilinson resolution
No full textGiven a smooth stacky Calabi-Yau hypersurface X in a weighted projective space, we consider the functor G which is the composition of the following two autoequivalences of D^b(X): the first one is induced by the spherical object O_X, while the second one is tensoring with O_X(1). The main result of the paper is that the composition of G with itself w times, where w is the sum of the weights of the weighted projective space, is isomorphic to the autoequivalence "shift by 2". The proof also involves the construction of a Beilinson type resolution of the diagonal for weighted projective spaces, viewed as smooth stacks
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