Let $C \subseteq \mathbb{A}_{\mathbb{C}}^2$ be a curve defined by a polynomial equation $f(x,y) = 0$, where $f \in \mathbb{Q}[x,y]$ is irreducible (and not necessarily non-singular). In the context I am interested in, application of Siegel's theorem, it just states “genus is positive'' without any further explanation all the time. I realized I don't have a good understanding of this (the thing I had in mind was the genus degree formula $(d-1)(d-2)/2$ but it is valid only if $C$ is nonsingular projective curve in $\mathbb{P}^2$…)
I would appreciate if someone could please clarify me the following questions.
-If it just says genus, is it arithmetic genus or geometric genus?
-How does one define a genus of an affine curve?
Any clarification is greatly appreciated.
ps as mentioned in the comment, couple places that mention genus of an affine curve:
Best Answer
The definition of geometric genus for a smooth projective geometrically irreducible curve $C$ over a field $k$ is $\dim_k H^0(C,\Omega_{C/k})$, and this is extended to curves which are not necessarily smooth or projective in the following manner: any finitely generated field extension of transcendence degree one $k\to K$ determines a regular projective curve over $k$ which is unique up to isomorphism, so we may define the geometric genus for any curve to be the geometric genus of a smooth projective model which is birational to our curve (NB: there's some trickiness here when you're dealing with a curve over an imperfect field which I am going to gloss over, see for instance 0BYE if you need this). This is the genus that is being referred to in your post.
As far as "which genus", arithmetic genus isn't typically defined for non-proper schemes: the definition of the arithmetic genus for a proper scheme $X$ over a field $k$ is $$(-1)^{\dim X}(\chi(\mathcal{O}_X)-1)=(-1)^{\dim X}\left(-1+\sum_{i=0} (-1)^i\dim_k H^i(X,\mathcal{O}_X)\right)$$ and if $X$ is not proper, there's no guarantee that $\dim_k H^i(X,\mathcal{O}_X)$ is finite. So if you're dealing with a non-proper curve and someone says genus, they can't mean arithmetic genus. Further, if $X$ is a smooth curve, the geometric genus and arithmetic genus are the same by Serre duality - so the only time you'd need to worry about the disagreement of these genera for curves is when you are dealing with a proper singular curve, and the author should specify here.