# Genus of a smooth projective curve

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).

• 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? – KReiser Jan 16 at 21:00
• I would like to see a proof using Riemann Hurwitz formula – Tommaso Scognamiglio Jan 16 at 21:44
• 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 $P^2$. – aginensky Jan 16 at 22:52
• I think this is basically a duplicated of this question. – André 3000 Jan 17 at 4:17
• @André3000 Tthe answer in there is just a broken link, it might be worth it having a written answer. – user347489 Jan 17 at 8:31