Geometry of the Discriminant Surface for Quadratic Forms

Details Geometry of the Discriminant Surface for Quadratic Forms Geometry of the Discriminant Surface for Quadratic Forms

2009-07-20

arxiv.org/abs/0907.3293v6

Mathematics

Algebraic Geometry

26 pages. Cites 9 references. A draft paper. Withdraw earlier versions. Changes since 2009: 1) computer algebra part removed,

Scientific paper

We investigate the manifold $\cal{M}$ of (real) quadratic forms in n > 1 variables having a multiple eigenvalue. In addition to known facts, we prove that 1) $\cal{M}$ is irreducible, 2) in the case of n = 3, scalar matrices and only them are singular points on $\cal{M}$. For $n = 3$, $\cal{M}$ is also described as the straight cylinder over $\cal{M}$$_0$, where $\cal{M}$$_0$ is the cone over the orbit of the diagonal matrix $\diag(1,1,-2)$ by the orthogonal changes of coordinates. We analyze certain properties of this orbit, which occurs a diffeomorphic image of the projective plane.

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