Riemann Roch: genus 3 curves

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This talk is about the Riemann-Roch theorem for genus 3 curves. We show that any such curve is either hyperelliptic or a nonsingular plane quartic. We find the Weierstrass points and the holomorphic 1-forms and the canonical divisors of these curves. Finally we give a brief description of the moduli space of genus 3 curves.
  1. Richard E Borcherds
  2. RIemann-Roch theorem
  3. genus 3 curve
  4. Weierstrass point
  5. Clifford line
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So hyperelliptic curves of genus 3 are sort of a degenerate case of quartic curves of genus 3 then.
While all genus 2 curves are hyperelliptic?
Do we have to work over the complex numbers for the theory (of algebraic geometry, algebraic topology and Riemann Roch) to work, or is it enough to work over an algebraically closed field (such as the infinite dimensional algebraic closure of the rationals, or of the p-adic numbers)? How does the theory work in characteristic p?

henrikljungstrand