# Algebraic geometry fun facts for the festivities: genera of complete intersection curves

Are you dreading the social responsibilities that come with this time of the year? You can spice up the conversations with your relatives and friends using the following fun facts in algebraic geometry. Whenever a conversational lull comes up, drop one of these and things will turn fun fun fun again!

## Genera of complete intersection curves

This MathOverflow question tells you an interesting fun fact: not all genera even arise as the genus of some complete intersection curve. Of course, complete intersections are quite special among projective varieties, no arguing about that. But that there are whole moduli spaces of $\mathcal{M}_g$ not containing a single complete intersection might be a bit puzzling at first.

Also, at some point the number of moduli for curves of some genus (i.e. $3g-3$) overtakes the number of moduli for complete intersection curves of the same genus, so you'd never expect to be able to describe generic curves of some genus as a complete intersection. But the point is that for some genera there are no complete intersection curves at all.

The reason is obvious once you think about it for a short while. The canonical bundle for a hypersurface $X$ of degree $d$ in $\mathbb{P}^n$ is $\mathcal{O}_X(-d)$, e.g. using the adjunction formula. Induction on the conormal exact sequence tells you that for a complete intersection of degree $(d_1,\ldots,d_k)$ we have $\omega_X\cong\mathcal{O}_X(\sum_{i=1}^kd_i-n-1)$ and hence computing its degree we get $$ g=1+\frac{1}{2}\left( \prod_{i=1}^{n-1}d_i \right)\left( \sum_{i=1}^{n-1}d_i - n - 1 \right) $$

Another way of getting there (it's just a different flavour of the same argument) is induction on the short exact sequence for a divisor and compute the first cohomology group of the structure sheaf.

Plugging in all possible values for the degrees and dimensions of the ambient projective space we get the following list of possible genera up to 100:

`0, 1, 3, 4, 5, 6, 9, 10, 13, 15, 16, 17, 19, 21, 25, 28, 31, 33, 36, 37, 41, 45, 46, 49, 51, 55, 61, 64, 65, 66, 73, 76, 78, 81, 85, 91, 97, 99`

So observe that the list of genera on MathOverflow contains a few mistakes: as Noam Elkies points out the value 2 does not appear (hyperelliptic curves are never complete intersections). And genus 6 appears as a plane degree 5 curve, but this value is omitted from the list. Also genus 12 does not appear, yet 13 does using a complete intersection of degree (3,2,2). You can check this easily using either Macaulay2 or my cohomology for complete intersections tool.

In case you want to toy around with it yourself, the code is available.