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I was reading wikipedia page on intersection theory, and it says that one can define self-intersection of a curve, and sometimes the number is negative. In the case there is a small deformation of the curve, say C' is a small deformation of C, one can define C.C=C.C', and this is quite intuitive. However, if the curve is an exceptional divisor of a blowup, there is no small deformation and I can't quite imagine geometrically how the self-intersection can be negative. Any idea?

Also, does anyone know a reference to Castelnuovo's contraction theorem that any (-1)-curve can be contracted?

best, minimax

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A good reference for Castelnuovo's result is Beauville's "Complex Algebraic Surfaces." You might find the following interesting as well:… – Andrew Dec 8 '12 at 6:44
Another reference would be Qing Liu's "Algebraic Geometry and Arithmetic Curves". – Thom Tyrrell Dec 8 '12 at 7:45

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