This may be a dumb question but is calculating the genus of a curve define over a finite field different than over $\mathbb{C}$. For example the following curve: $y^8 + y +x^{12} + x^5$ is genus 14 over GF(8) according to Magma. This curve is non-singular (I believe?) and therefore if define over $\mathbb{C}$ the genus would follow a very simple degree formula but in this case does not so something is happening because of the field of definition.... What is it?

  • $\begingroup$ What degree formula are you talking about? Maybe you are assuming that this curve is Hyperelliptic? $\endgroup$ – Ravi May 27 '16 at 16:12
  • $\begingroup$ Degree 8 i think? $\endgroup$ – Tom Lewia May 27 '16 at 16:16
  • $\begingroup$ What genus are you talking about? Do you mean the genus of the closure of the curve in $\mathbb P^2$? In that case, the closure is singular. $\endgroup$ – Fredrik Meyer May 27 '16 at 17:49

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