We know that blowing up a point on a surface produces a $(-1)$ curve. Is there any such standard techniques to produce $(-2)$ curves in a smooth surface?
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6$\begingroup$ Dear Jana, the quick answer is yes: on any surface, blow up a smooth point, then blow up a point on the exceptional curve. The proper transform of the first exceptional curve will then be a (-2)-curve on the second blow-up. Is that the kind of answer you're looking for? $\endgroup$– user5117Commented Mar 19, 2013 at 20:25
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2 Answers
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In case you want a curve of arbitrary genus with arbitrary negative self-intersection, you can do this: Let $C$ be a smooth projective genus $g$ curve on a smooth surface. Suppose $C^2=n$ and take $m$ (pairwise) different points on $C$ and blow them up. The strict transform of $C$ is isomorphic to $C$, so it has genus $g$, and has self-intersection $n-m$. In other words, you can achieve arbitrary combinations of genus and negative-self-intersection.
answered Mar 20, 2013 at 4:36
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$\begingroup$ You dont need an infinite field in fact, you just need a smooth point defined on your field. You can take a smooth point on the curve, then blow-up the point infinitely near to it corresponding to the tangent direction and so on. $\endgroup$ Commented Mar 20, 2013 at 6:01
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$\begingroup$ Jérémy: you're right. I just wanted to make it a one-step process.... $\endgroup$ Commented Mar 20, 2013 at 6:47
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Here is something that might help answer your question:
Blowing-up rational double points on normal singular surfaces produces $(-2)$-curves of genus zero. Conversely Artin [1, Thm. 2.7] showed that (under suitable conditions) every such $(-2)$-curve of genus zero arises in this way.
[1] Artin - Some numerical criterion for contractability of curves on surfaces.
answered Mar 19, 2013 at 20:27