---
tag: CQJ2E2ARRF
kind: concept
name: Characteristic variety
msc: [35A30, 35A27]
sources:
  - url: https://en.wikipedia.org/wiki/Characteristic_variety
---

## Summary

The characteristic variety of a linear differential operator is the set of points and nonzero covectors where its [[principal symbol|F31T650DKY]] vanishes.
A hypersurface whose normal never lies in this set is called noncharacteristic, and the Cauchy problem with data on such a hypersurface is well behaved.
In algebraic analysis the same idea defines the characteristic variety of a D-module.
See the [[definition|JMJEVS7S1V]].

## Explanations

### L1 human

The characteristic variety lists the directions in which an operator fails to control the highest derivatives.
Along such directions, data on a surface do not determine how the solution leaves the surface.
For the [[wave equation|5EHD2XBQDG]] $\partial_t^2 u - \partial_x^2 u$, the symbol $\tau^2 - \xi^2$ vanishes when $\tau = \pm \xi$.
The corresponding curves $x \pm t = \text{const}$ are the light rays along which waves travel.