Determinant variety

The set of matrices $ D _ {t} ( d , n ) $ of dimension $ d \times n $ and of a rank lower than $ t $, with the structure of an algebraic variety. Let $ J _ {t} ( d , n ) $ be the ideal in the ring of polynomials

$$ k \left [ ( T _ {ij} ) _ {\begin{array} {l} 1 \leq i \leq d \\ 1 \leq j\leq n \end{array} } \right ] , $$

with coefficients in a field $ k $, generated by the $ t $- order minors of the matrix of dimension $ d \times n $ constituted by the variables $ T _ {ij} $( a determinant ideal). The set of zeros of the ideal $ J _ {t} ( d , n ) $ in the affine space $ A ^ {dn} = { \mathop{\rm Spec} } ( k [ ( T _ {ij} ) ] ) $ is known as the determinant variety and is denoted by $ D _ {t} ( d , n ) $. For any commutative $ k $- algebra $ k ^ \prime $ the set of $ k ^ \prime $- points of the determinant variety $ D _ {t} ( d , n ) $ coincides, in a natural manner, with the set of matrices of dimension $ d \times n $ and rank $ < t $ with coefficients in $ k ^ \prime $.

The following are special cases of determinant varieties: $ D _ {d} ( d , n ) $ is the hypersurface in $ A ^ {d ^ {2} } $ defined by the vanishing of the determinant of a square matrix of dimension $ d $ consisting of independent variables (a determinant hypersurface); $ D _ {2} ( d , n ) $ is an affine cone for the image of the Segre imbedding

$$ P ^ {d-1} \times P ^ {n-1} \rightarrow P ^ {dn-1} $$

of the product of projective spaces [2].

Determinant varieties have the following properties: $ D _ {t} ( d , n ) $ is irreducible, reduced (i.e. the ideal $ J _ {t} ( d , n ) $ is simple), is a Cohen–Macaulay variety (cf. Cohen–Macaulay ring), is normal, and the dimension of $ D _ {t} ( d , n ) $ is equal to $ ( t - 1 ) ( n + d - 1 ) $[1], [2]. $ D _ {t} ( d , n ) $ is a Gorenstein scheme if and only if $ t = 1 $ or $ d = n $ (cf. Gorenstein ring) [5]. Determinant varieties are closely connected with Schubert varieties of a Grassmann manifold (cf. Schubert variety).

References

[1]	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
[2]	S. Kleiman, J. Landolfi, "Geometry and deformation of special Schubert varieties" Compositio Math. , 23 (1971) pp. 407–434
[3]	D. Laksov, "Deformation of determinantal schemes" Compositio Math. , 30 (1975) pp. 273–292
[4]	C. Musili, "Some properties of Schubert varieties" J. Indian Math. Soc. , 38 (1974) pp. 131–145
[5]	T. Svanes, "Coherent cohomology on Schubert subschemes of flag schemes and applications" Adv. in Math. , 14 (1974) pp. 369–453

Comments

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

References

[a1]	T.G. Room, "Geometry of determinantal loci" , Cambridge Univ. Press (1938)