# Algebraic geometry fun facts for the festivities: Enriques surfaces are never complete intersections

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!

## Enriques surfaces are never complete intersections

Whilst preparing the previous fun facts for the festivities post I came across this MathOverflow question. So continuing on the theme of what can (not) be realised as a complete intersection we now present you the Enriques surfaces. As Daniel Loughran explains, because the canonical bundle on an Enriques surface is torsion it cannot be the restriction of a line bundle on some hypothetical ambient projective space in which it is a complete intersection (which it should be by iterating the adjunction formula), because $\mathrm{Pic}(\mathbb{P}^n)=\mathbb{Z}$ injects into the Picard group of the Enriques surface.

These guys are also closely related to K3 surfaces, for which there *are* complete intersection examples. The most famous one being quartic surfaces, the rather short list continues with the intersection of a quadric and a cubic in $\mathbb{P}^4$ and the intersection of three quadrics in $\mathbb{P}^5$.