# Counting Lines and Conics on a Surface

In fact I am trying to find the conditions on curves to be embedded in surfaces (non ruled). Apart from reading the books by Mumford and Kollar I found the following paper useful.

http://www.kurims.kyoto-u.ac.jp/~prims/pdf/45-3/45-3-25.pdf

The paper is well written so I don't think I need to write the techniques of this paper. Now I am looking for references that describe other kind of techniques available in this area.

Thanks.

-
@Algebraic geometer: You probably have something in mind. Why not ask the real question you are interested in? – Sándor Kovács Dec 26 '11 at 21:06
I certainly agree with @Sandor re asking the real question. – Igor Rivin Dec 26 '11 at 21:18
@S'andor -- This is fun! Do you know the question the OP is interested in? Are you willing to share with the rest of the class? – Jason Starr Dec 28 '11 at 4:35
@S. Carnahan: I am sorry but I don't consider my writing abilities better than Yoichi Miyaoka, and I don't want to type most of the same paper here again. One way of putting my query is as follows: What are the techniques known related to conics (lines etc ) on an irreducible non-singular surface (non-ruled)? For example, simply, consider the case of number of conics on on a quintic surface?(the max you can get is 4 conics on a quintic surface, you can get 5 conics but then the surface is reducible). – Algebraic geometer Dec 29 '11 at 15:38
@Algebraic geometer: There is an upper bound on the number of lines on a smooth degree $d$ surface in $\mathbb{P}^3$, namely $11d^2-28d+12$. This is the correct order of magnitude, as the Fermat surface contains $3d^2$ lines. You might want to look at my appendix to the following article arxiv.org/abs/math/0505186. – Jason Starr Jan 3 '12 at 4:39