# Geometric Intuition for the Genus of a Curve

More specifically, the genus of a curve in projective space. And by a curve, I mean a complete nonsingular curve over an algebraically closed field k, or in other words, an integral scheme of dimension 1, proper over k, all of whose local rings are regular.

If it would be possible, could you explain it at different levels? So like layman, then undergrad, then basic grad, and then a really formal grad level explaination, or at least an intuitive remark...

Anything you can contribute would be greatly appreciated <3

• The basic historical intuition is that a complex curve is a real surface so it has a topological genus as in "the number oh holes". Apr 6, 2020 at 8:32
• @CaptainLama Well yeah, I know about topological genus, but what about the arithmetic/geometric genus of a curve? Apr 6, 2020 at 8:43
• All three notions are the same in the case of a smooth projective curve over $\Bbb C$. Arithmetic/geometric genus are the same in the smooth projective case over arbitrary fields (if $\mathcal{O}_X(X)=k$ and our curve is geometrically irreducible), and they can differ once your curve starts becoming singular or otherwise less well-behaved, though there are ways to account for the differences. I'll see if I can't develop this in to a better answer later, though others are certainly welcome to use what I've written without waiting for me. Apr 6, 2020 at 9:38
• The geometric genus is the number of linearly independent regular differentials on the curve. Matt Emerton gave a good answer on the topic of genus and divisors here. Apr 6, 2020 at 15:34