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This semester I took a lecture on Riemann surfaces. The professor proved the Riemann-Roch theorem (stated below). As an application of it, he proved elementary results, we did earlier in the course anyway, e.g. that the Riemann-sphere has genus $0$.

My question is: What are the (less elementary) applications and implications of the Riemann-Roch theorem in the form below?

Theorem: Let $X$ be a compact Riemann-surface, $D$ a divisor of $X$. Then the Cech-cohomology groups $H^0 (X, \mathcal{O}_D)$ and $H^1(X, \mathcal{O}_D)$ are finite-dimensional vector-spaces and $$\dim H^0 (X, \mathcal{O}_D) - \dim H^1(X, \mathcal{O}_D) = 1 - g + \deg D$$ where $\mathcal{O}_D$ is the sheaf with $\mathcal{O}_D(U) := \{ f \text{ meromorphic on } X : ord_x(f) \geq -D(x), \forall x \in U \}$ and the natural restriction maps, and $g$ is the genus of $X$.

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  • $\begingroup$ I think already the ability to say that a bundle is ample, basepoint free, very ample, etc based on the degree is very neat. If you want "nontrivial" look at the first volume of Arbarello-Cornalba-Griffiths-Harris, Geometry of Algebraic Curves. $\endgroup$ – Hoot Jun 22 '16 at 20:12
  • $\begingroup$ You can use it to prove that the dimension of the space of complex structures on a Riemann surface of genus $g$ is $6g-6$ (when $g>1$), but I'm not expert enough in the algebraic way of thinking of things to know how to prove this there. $\endgroup$ – user98602 Jun 22 '16 at 20:40
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    $\begingroup$ My copy of Farkas and Kra is all-too-fresh-looking, i.e. I don't know this topic well. But Chapter III.5 of that book contains some applications, such as the Weirstrass Gap Theorem. $\endgroup$ – Lee Mosher Jun 22 '16 at 21:01
  • $\begingroup$ More elementary than Arbarello-Cornalba-G-H would be something like Griffiths's Introduction to Algebraic Curves. $\endgroup$ – Ted Shifrin Jun 28 '16 at 0:30

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