Generalizing BCnrd's example, almost all surfaces S of degree d ≥ 4 in P^3 have Picard group = Z, hence every curve on a general S is a complete intersection with another surface. Thus any two curves on S intersect. This is the theorem of Noether-Lefschetz.

http://www.springerlink.com/content/t754510m417u0712/

Generalizing BCnrd's example, almost all surfaces S of degree d ≥ 4 in P^3 have Picard group = Z, hence every curve on a general S is a complete intersection with another surface. Thus any two curves on S intersect. This is the theorem of Noether-Lefschetz.

http://www.math.umsl.edu/~girivaru/dcg.pdf