Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field and consider the intersection pairing on the surface $X \times X$. I remember hearing that $\Delta^2 = 2-2g$: how does one prove this? I understand that the self-intersection is defined by intersecting $\Delta$ with a general divisor linearly equivalent to $\Delta$, but I'm in the dark about how to compute such things.
Also, what are $(X \times \{ * \})^2$ and $(\{ * \}\times X)^2$? My intuition suggests that these are both $0$, but again I don't know how to compute.