Abstract blowing down

Abstract

Assume that X X is a surface over an algebraically closed field k k . Let X ~ \tilde {X} be obtained from X X by blowing up a smooth point and let L L be the exceptional curve. Let coh ⁡ ( X ) \operatorname {coh}(X) be the category of coherent sheaves on X X . In this note we show how to recover coh ⁡ ( X ) \operatorname {coh}({X}) from coh ⁡ ( X ~ ) \operatorname {coh}(\tilde {X}) , if we know the object O L ( L ) \mathcal {O}_L(L) .