Geodesic
Derived Categories of Fano Threefolds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Multidimensional algebraic geometry, Tome 264 (2009), pp. 116-128

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We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the appendix we discuss how the ring of algebraic cycles of a smooth projective variety is related to the Grothendieck group of its derived category. 
@article{TM_2009_264_a13,
     author = {A. G. Kuznetsov},
     title = {Derived {Categories} of {Fano} {Threefolds}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {116--128},
     publisher = {mathdoc},
     volume = {264},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2009_264_a13/}
}
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A. G. Kuznetsov. Derived Categories of Fano Threefolds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Multidimensional algebraic geometry, Tome 264 (2009), pp. 116-128. http://geodesic.mathdoc.fr/item/TM_2009_264_a13/