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Some (probably very easy) questions on intersection theory on surfaces...

Say $S$ is a smooth projective surface over $\mathbb{C}$ with canonical divisor $K_S$.

  • If $S$ is not ruled and $H$ is a hyperplane section (for an arbitrary embedding), do we always have that $K_S \cdot H > 0$? I know that $K_S \cdot H \geq 0$, but why would $K_S \cdot H = 0$ be impossible? EDIT: OK, this can happen if $K_S = 0$, of course (see QiL's answer). But if I moreover assume that $K_S^2 > 0$, can we still have $K_S \cdot H = 0$?

  • If $D \cdot H < 0$ for some divisor $D$ and some hyperplane section $H$, does it follow that $H^0(D,\mathcal{O}_S(D)) = 0$? It seems reasonable but also too simple, so I'm not so sure.

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  1. It can happen that $K_S=0$ (i.e. $S$ is a K3 surface), then $K_S\cdot H=0$.

  2. If $H^0(D, O_S(D))\ne 0$, then up to linear equivalence, $D\ge 0$, so $D\cdot H\ge 0$ because $H$ is ample.

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    @Evariste, then K_S\cdot H>0. See Hodge index theorem (Hartshorne, V.1.9). For D\cdot H>0: there is a hyplane $H'$ which doesn't contain $D$. So D\cdot H=D\cdot H'>0. See also Hartshorne V.1.10 for the inverse implication.2012-05-31