---
tag: Z014Q8KPDD
kind: concept
name: Schubert variety
msc: [14M15, 14N15]
sources:
  - url: https://en.wikipedia.org/wiki/Schubert_variety
same_as:
  wikidata: Q7432938
  wikipedia: Schubert variety
---

## Summary

A Schubert variety is a subvariety of a [[Grassmannian|CYPA882RB9]] or a [[flag variety|NHRWCQQKTH]] defined by conditions on how a subspace meets a fixed flag.
In a Grassmannian they are indexed by partitions that fit in a rectangle.
Their classes form a basis of the cohomology ring, and computing products of these classes is [[Schubert calculus|FWGHMADJR4]].
Schubert varieties are usually singular, and their singularities are studied through combinatorics.

## Explanations

### L1 human

Fix a line $\ell$ in three-dimensional projective space.
The lines that meet $\ell$ form a three-dimensional family inside the four-dimensional space of all lines.
This is a Schubert variety: it is defined by one incidence condition with a fixed subspace.
Imposing stronger conditions, such as lying in a fixed plane or passing through a fixed point, gives smaller Schubert varieties.