I read the following statement:
Let $\mathcal L$ be a line bundle with $(\mathcal L.\mathcal L)>0$ on surface. If $(\mathcal L'.\mathcal L)=0$, then $(\mathcal L'.\mathcal L')\leq0$.
This looks very like the Hodge index theorem, except that the condition $\mathcal L$ being ample is weaken by $(\mathcal L.\mathcal L)>0$. I followed the proof of Hodge index theorem and I think the condition of ampleness is essential (I followed the Hartshorne's book, whose proof is based on Lemma 5.1.7, which I cannot see still holds true if ampleness is replaced by $(\mathcal L.\mathcal L)>0$).
So, how can I prove this statement? Is there any trick to avoid the ampleness in the proof?
Thanks in advance.