4.4 Blowing Down

Suppose \(X\) is a surface and \(E = \cup _{i=1}^r E_i\) is a connected curve with pairwise distinct integral components \(E_i\). Then the negative definiteness of the matrix \(E_i\cdot E_j\) is a necessary condition to have a proper birational morphism \(f:X\to Y\) from \(X\) to a normal surface \(Y\) contracting \(E\) to a point. In the case of algebraic spaces, it is also sufficient by a result of Grauert. However, an example of Nagata shows that it is not sufficient if we want \(Y\) to be a scheme.