---
tag: AV2ZAAZXN9
kind: concept
name: Del Pezzo surface
msc: [14J26, 14J45]
specializes: [FMHA649HJ9, 55A4A53TRF]
sources:
  - url: https://en.wikipedia.org/wiki/Del_Pezzo_surface
same_as:
  wikidata: Q2067112
  wikipedia: Del Pezzo surface
---

## Summary

A del Pezzo surface is a smooth projective [[algebraic surface|KFTECAATKG]] whose anticanonical divisor is ample.
These are exactly the [[Fano varieties|FMHA649HJ9]] of dimension two.
Over an algebraically closed field they are $\mathbb{P}^1 \times \mathbb{P}^1$ and the blow-ups of the plane in at most eight points in general position.
So they are [[rational surfaces|55A4A53TRF]].
Their degree $K^2$ runs from $1$ to $9$, and the curves of self-intersection $-1$ on them are governed by root systems of type $E$.

## Explanations

### L1 human

Del Pezzo surfaces are the most positively curved surfaces in the algebraic sense.
One gets them by taking the projective plane and blowing up a few points in sufficiently general position.
Each blown-up point is replaced by a line, and new lines appear from the lines and conics through the chosen points.
The smooth cubic surfaces in space, with their $27$ lines, are the del Pezzo surfaces of degree $3$.