View source for Determinant variety

The set of matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d0314201.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d0314202.png" /> and of a rank lower than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d0314203.png" />, with the structure of an algebraic variety. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d0314204.png" /> be the ideal in the ring of polynomials

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with coefficients in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d0314206.png" />, generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d0314207.png" />-order minors of the matrix of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d0314208.png" /> constituted by the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d0314209.png" /> (a determinant ideal). The set of zeros of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142010.png" /> in the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142011.png" /> is known as the determinant variety and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142012.png" />. For any commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142013.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142014.png" /> the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142015.png" />-points of the determinant variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142016.png" /> coincides, in a natural manner, with the set of matrices of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142017.png" /> and rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142018.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142019.png" />.

The following are special cases of determinant varieties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142020.png" /> is the hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142021.png" /> defined by the vanishing of the determinant of a square matrix of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142022.png" /> consisting of independent variables (a determinant hypersurface); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142023.png" /> is an affine cone for the image of the Segre imbedding

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of the product of projective spaces [[#References|[2]]].

Determinant varieties have the following properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142025.png" /> is irreducible, reduced (i.e. the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142026.png" /> is simple), is a Cohen–Macaulay variety (cf. [[Cohen–Macaulay ring|Cohen–Macaulay ring]]), is normal, and the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142027.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142028.png" /> [[#References|[1]]], [[#References|[2]]]. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142029.png" /> is a Gorenstein scheme if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142030.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031420/d03142031.png" /> (cf. [[Gorenstein ring|Gorenstein ring]]) [[#References|[5]]]. Determinant varieties are closely connected with Schubert varieties of a Grassmann manifold (cf. [[Schubert variety|Schubert variety]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Hochster,   J. Eagon,   "Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci"  ''Amer. J. Math.'' , '''93''' :  4  (1971)  pp. 1020–1058</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kleiman,   J. Landolfi,   "Geometry and deformation of special Schubert varieties"  ''Compositio Math.'' , '''23'''  (1971)  pp. 407–434</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Laksov,   "Deformation of determinantal schemes"  ''Compositio Math.'' , '''30'''  (1975)  pp. 273–292</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Musili,   "Some properties of Schubert varieties"  ''J. Indian Math. Soc.'' , '''38'''  (1974)  pp. 131–145</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T. Svanes,   "Coherent cohomology on Schubert subschemes of flag schemes and applications"  ''Adv. in Math.'' , '''14'''  (1974)  pp. 369–453</TD></TR></table>



====Comments====
Many geometrical properties of determinant varieties can be found in [[#References|[a1]]]. Instead of determinant variety, etc. one also speaks of determinantal variety, etc.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.G. Room,   "Geometry of determinantal loci" , Cambridge Univ. Press  (1938)</TD></TR></table>