Derived categories of nodal del Pezzo threefolds

Abstract

We give a complete answer for the existence of Kawamata type semiorthogonal decompositions of derived categories of nodal del Pezzo threefolds. More precisely, we show that nodal del Pezzo threefolds of degree $1\le d\le 4$ have no Kawamata type decomposition and that all nodal del Pezzo threefolds of degree $5$ admit a Kawamata decomposition. For the proof we go through the classification of singular del Pezzo threefolds, compute divisor class groups of nodal del Pezzo threefolds of small degree and use projection from a line to construct Kawamata semiorthogonal decompositions for the degree $5$ case. An analogous decomposition of the nodal del Pezzo threefold of degree $6$ has been recently constructed by Kawamata. Our construction of the Kawamata decomposition for a singular del Pezzo threefold of degree $5$ fits into a family of semiorthogonal decompositions (which we call a relative tilting decomposition) interpolating between a Kawamata decomposition on a singular fiber and a full exceptional collection on the smooth fibers.

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Cited by in corpus (2)

• Categorical absorptions of singularities and degenerations
• Nodal quintic del Pezzo threefolds and their derived categories