MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Fix a flag of subspaces V1 in V2 in V3, etc. all in Cn.

Consider the space of pairs of commuting linear transformations A and B such that:
A preserves the flag (i.e. A(Vi) is in Vi), and
B strictly preserves the flag (i.e. B(Vi) is in Vi-1).

Does anyone know anything about this space? Is there any literature on it? Is it smooth?

Just as an example: if the flag is trivial, B=0 and A can be anything. So that's smooth.

share|cite|improve this question

for the case where n = 2 and the complete flag the variety in question is isomorphic to A^2 x Spec C[x,y]/(xy). so the space is not nec. smooth.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.