Abstract. We provide a categorical framework for recent results of Markus Perling on combinatorics of exceptional collections on numerically rational surfaces. Using it we simplify and generalize some of Perling's results as well as Vial's criterion for existence of a numerical exceptional collection.

We show that the bounded derived category of coherent sheaves on a general Enriques surface can be realized as a semiorthogonal component in the derived category of a smooth Fano variety with diagonal Hodge diamond.If a smooth projective variety X over the field C of complex numbers has a full exceptional collection, then its Hodge diamond is diagonal, i.e., h p,q (X) = 0 for p = q.It is natural to ask whether the converse is true. A simple counterexample to this naive question is provided by an Enriques surface S -its Hodge diamond looks like

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Мы показываем, что ограниченная производная категория когерентных пучков на общей поверхности Энриквеса может быть реализована как полуортогональная компонента в производной категории гладкого многообразия Фано с диагональным ромбом Ходжа. Библиография: 14 наименований.