Let \(X\) be a smooth surface, separated and of finite type over an algebraically closed field \(k\), and let \(E \subseteq X\) be an integral curve that is proper over \(k\). There exist a smooth surface \(Y\) over \(k\), a point \(p \in Y\) and a morphism \(\sigma \colon X \to Y\) that is the blowup of \(Y\) at \(p\) with exceptional curve \(E\) if and only if \(E \cong \PP^1\) and \(E^2 = -1\). Such a curve is a \((-1)\)-curve, or exceptional curve of the first kind. If \(X\) is projective, then \(Y\) is projective [@Har10a, Theorem V.5.7].
This converts a numerical condition into a birational morphism, and it is what makes the minimal model program work in dimension two: contract \((-1)\)-curves until none remain, and the result is a minimal surface. Since each contraction raises \(K^2\) by one and \(\rho\) drops by one, the process terminates.
By adjunction, \(E \cong \PP^1\) with \(E^2 = -1\) is equivalent to \(E^2 = -1\) and \(K \cdot E = -1\), which is the form the criterion is usually applied in.
A surface can have infinitely many \((-1)\)-curves: the blowup of \(\PP^2\) at \(9\) points in general position, in the sense that no three of the points are collinear after any finite sequence of quadratic transformations centred at three of them, has infinitely many (P-AGH5415GENPOSITION, part (e)), while for \(r = 7\) and \(r = 8\) points in general position there are exactly \(56\) and \(240\) (part (d)). The minimal model is then not unique in general — for rational surfaces one can reach both \(\PP^2\) and the Hirzebruch surfaces — which is exactly the dimension-two failure that makes the classification of rational and ruled surfaces a separate theorem.
Dependencies
FE-SRFBLOWWhat a blowup does to $\Pic$, to $K$, and to the intersection formuses
Backlinks
D-SCHBLOWUPBlowing up a scheme along a closed subscheme, and blowing downrelated-toT-CRVMINMODEvery curve has a unique smooth projective modelrelated-toT-SRFKODKodaira dimension and the Enriques classification of surfacesusesD-SRFRULEDRational and ruled surfaces, and the Hirzebruch surfacesusesT-SRFZMTZariski's main theorem, and the factorization of birational maps of surfacesuses