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?

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 blowup. Is that the kind of answer you're looking for? $\endgroup$ – user5117 Mar 19 '13 at 20:25
In case you want a curve of arbitrary genus with arbitrary negative selfintersection, 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 selfintersection $nm$. In other words, you can achieve arbitrary combinations of genus and negativeselfintersection.

$\begingroup$ 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 blowup the point infinitely near to it corresponding to the tangent direction and so on. $\endgroup$ – Jérémy Blanc Mar 20 '13 at 6:01

$\begingroup$ Jérémy: you're right. I just wanted to make it a onestep process.... $\endgroup$ – Sándor Kovács Mar 20 '13 at 6:47
Here is something that might help answer your question:
Blowingup 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.