Let $X$ be a $K3$ surface and $\sigma$ an antisymplectic involution on $X$ and so $X$ is algebraic. 1.Why the signature of $Pic(X)$ is $(1,\rho 1)$? (This is well known but I cant find any direct proof) 2. Lets assume the set of fixed point set of $\sigma$ which we denote by $Fix(\sigma)$ contains a smooth curve say $D_g$ of genus $g\geq 2$, let's assume we have other curve in $Fix(\sigma)$ which is elliptic say $D$, i.e., we should have $D^2=0$ and $D.D_g=0$ then why this is a contrary to the fact that the signature of $Pic(X)$ is $(1,\rho 1)$. 3. How do we see curves inside $Pic(X)$? I mean curves are inside $X$ and $Pic(X)$ is a lattice!
