This is a problem I encountered when reading Algebraic Curves and Riemann Surfaces by Rick Miranda:

Show that if $X$ is a hyperelliptic curve with genus $g$ defined by $y^2=h(x)$, then the space $\Omega^1(X)=\{p(x)\frac{dx}{y}|\deg(p)\leq g-1\}$.

Any thoughts?

  • $\begingroup$ any thoughts? ${}$ $\endgroup$ – user347489 Apr 28 '18 at 22:18
  • 1
    $\begingroup$ You need to prove these are all actually (linearly independent) holomorphic differentials. The definition of genus $g$ is $\dim H^0(X,\Omega^1) = g$. $\endgroup$ – Ted Shifrin Apr 28 '18 at 22:48

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