Let $X$ be a smooth projective threefold over the space of complex numbers. Let $L$ be a line bundle on $X$ such that $D_1,D_2\in |L|$ are linearly equivalent disjoint divisors.
Then is it true that $L^2=L^3=0$? Is it possible to have another line bundle $H$ such that $L^2.H>0$?
This seems like a silly question to me. But I am learning about nef line bundles and playing around with some examples to get a feel for it. Any intuition/help will be very useful.