n-blocks collections on Fano manifolds and sheaves with regularity −∞

Abstract

Let X be a smooth Fano manifold equipped with a “ nice ” n-blocks collection in the sense of [3] and F a coherent sheaf on X . Assume that X is Fano and that all blocks are coherent sheaves. Here we prove that F has regularity −∞ in the sense of [3] if Supp(F) is finite, the converse being true under mild assumptions. The corresponding result is also true when X has a geometric collection in the sense of [2].

Keywords

• m-block collection
• Fano manifold
• Coherent sheaves on a Fano manifold
• Order of regularity