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Jul 10 at 8:59 comment added Jason Starr An alternative argument is to use the asymptotics of (complex analytic) Riemann-Roch for the integer-valued function $\chi(S,\mathcal{O}_S(nD))$ of $n$, where $D$ is an effective divisor whose support is contained in $C$. The leading term is the self-intersection of $D$ on $S$. However, it is straightforward to see that both $h^0$ and $h^2$ do not grow more than linearly in $n$. Thus, the self-intersection must be nonpositive.
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