Abstract blowing down

Abstract

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