I was trying to prove that the genus of a smooth projective complex curve $F=0$ of degree $d$ is $(d-1)(d-2)/2$.

My attempt was to take the standard projection $$\pi:\mathbb{P}^2 \to \mathbb{P}^1$$ $$[x ;y ; z] \to [x ; z]$$ which has degree $d$ and then apply Riemann Hurwitz formula.

The problem is that I do not know how to explicitly write the multiplictity of this map at the ramification points(which are the ones for which $\dfrac{\partial F}{\partial y}=0$). In particular, I would like to say that the multiplicity of the map $\pi $ in a point $x \in \mathbb{P}^2$ is $I\left(x,F \cap \dfrac{\partial F}{\partial y}+1\right)$, but I'm stuck (the last I is the intersection multiplicity of the two algebraic curves).

  • $\begingroup$ Are you asking about how to finish this particular proof using the Riemann-Hurwitz formula, or are you interested in alternate ways to prove this statement? $\endgroup$ – KReiser Jan 16 at 21:00
  • $\begingroup$ I would like to see a proof using Riemann Hurwitz formula $\endgroup$ – Tommaso Scognamiglio Jan 16 at 21:44
  • $\begingroup$ Is it okay to use adjunction + cohomology ? Then $K_C = \mathcal{O}_C(n-3) $. Since the curve is defined by a single equation of degree d it's easy to see that $K_C$ has exactly the same global sections as on $P^2$. $\endgroup$ – aginensky Jan 16 at 22:52
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    $\begingroup$ I think this is basically a duplicated of this question. $\endgroup$ – André 3000 Jan 17 at 4:17
  • $\begingroup$ @André3000 Tthe answer in there is just a broken link, it might be worth it having a written answer. $\endgroup$ – user347489 Jan 17 at 8:31

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