Note on the dimension of certain algebraic sets of matrices

Details Note on the dimension of certain algebraic sets of matrices Note on the dimension of certain algebraic sets of matrices

2012-01-11

arxiv.org/abs/1201.2217v1

Mathematics

Algebraic Geometry

8 pages, some figures

Scientific paper

In this short note we prove a lemma about the dimension of certain algebraic sets of matrices. This result is needed in our paper arXiv:1201.1672. The result presented here has also applications in other situations and so it should appear as part of a larger work. The statement of the lemma goes as follows: Suppose $X$ is a nonempty algebraically closed subset of the affine space of $n \times m$ complex matrices. Suppose that $X$ is column-invariant (i.e., belongingness to $X$ depends only on the column space of the matrix). Suppose $E$ is a vector subspace of $\mathbb{C}^n$ that is not contained in the column space of any matrix in $X$. Then $codim(X) \geq m + 1 - dim(E)$. The proof is simple and relies on intersection theory of the grassmannians ("Schubert calculus").

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