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Rigid and exceptional vector bundles and sheaves on a Fano variety. (English) Zbl 0855.14001
Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 697-705 (1995).
A vector bundle \(E\) over an algebraic variety \(X\) is called “rigid” when \(\text{Ext}^1 (E, E) = 0\); this technical condition is in fact related to have \(E\) isolated in its moduli space. \(E\) is called “exceptional” when \(\text{Ext}^i (E,E) = 0\) for all \(i > 0\) and \(\operatorname{Hom} (E, E) = K\). When \(X\) is a Del Pezzo surface, then it turns out that \(E\) is exceptional if and only if it is stable and rigid.
The author provides a survey on recent results about the classification of exceptional vector bundles over Del Pezzo surfaces. Several open questions and conjectures are presented and discussed in the paper.
For the entire collection see [Zbl 0829.00014].

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J45 Fano varieties
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli