Geometry of the Discriminant Surface for Quadratic Forms

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Mechveliani, Sergei D.
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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.
Comment: 26 pages. Cites 9 references. A draft paper. Withdraw earlier versions. Changes since 2009: 1) computer algebra part removed, 2) results from Arnold's book referenced, 3) many points canceled, many points improved
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Mathematics - Algebraic Geometry
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