ag.algebraic geometry - Dimension of H^0(S,O_{S}(-C))

I'm not sure about the general case, but this is at least true if the divisor associated to the curve C is ample:

In this case the line bundle L associated to the divisor is ample, so the Kodaira-Nakano vanishing theorem applies. By Serre duality we get

$$0 = H^{2,1}(S,L) = H^1(S, Omega^2 otimes L) = H^1(S,L*)^* = H^1(S,O(-C))^*$$

as the dual of $L$ is the line bundle associated to the divisor $-C$. Remember that $mathcal I_C = O(-C)$, then the long exact sequence associated to

$$0 to O_S(-C) to O_S to O_C to O$$

gives the exact sequence

$$0 to H^0(S,O_S(-C)) to mathbb C to mathbb C to 0$$

as both $O_S$ and $O_C$ only have constant global sections. The vanishing of $H^0$ follows.

The general case would hold in the same way if $H^1(S,O(-C)) = H^1(S,mathcal I_C) = 0$ for any smooth curve $C$ in $S$.

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