# signature of picard lattice

Let's assume $X$ is a K3 surface. Why the signature of the picard lattice is $(1,\rho(X)-1)$? We know that (by Hodge index theorem) if we have a curve $D$ in $X$ with self intersection $D^2>0$ then any other curve $D_1$ with $D.D_1=0$ should have $D_1^2\leq 0$. Why $D_1^2=0$ can not happen? If $D_1^2=0$ what happens to the signature of Picard lattice?

-