---
tag: SV4CK2XP8M
kind: concept
name: Surface of general type
msc: [14J29]
specializes: [KFTECAATKG]
sources:
  - url: https://en.wikipedia.org/wiki/Surface_of_general_type
same_as:
  wikidata: Q16766069
  wikipedia: Surface of general type
---

## Summary

A surface of general type is an [[algebraic surface|KFTECAATKG]] of [[Kodaira dimension|ZW00Q0SAGR]] $2$, the largest possible value.
Its pluricanonical systems $\abs{nK}$ grow as fast as possible, quadratically in $n$.
Most surfaces are of this type, for example all smooth surfaces of degree at least $5$ in $\mathbb{P}^3$.
Unlike the other classes in the Enriques–Kodaira classification, they are not classified in detail.
Their study centres on inequalities between invariants and on their moduli spaces.

## Explanations

### L1 human

Surfaces of general type are the two-dimensional analogue of curves of genus at least two.
They are, in a loose sense, negatively curved and rigid, with many differential forms.
A surface given by a random polynomial equation of high degree in three-space is of general type.