Weak Del Pezzo Surfaces
A weak del Pezzo surface is a complete non-singular surface with anticanonical bundle that is nef and big.
The blowdown of any (−1)-curve on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 more. The blowup of any point on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 less, provided that the point does not lie on a −2-curve and the degree is greater than 1.
Any curve on a weak del Pezzo surface has self intersection number at least −2. The number of curves with self intersection number −2 is at most 9−d, and the number of curves with self intersection number −1 is finite.
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