In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, determinantal varieties are spaces of matrices with a given upper bound on their
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
s. Their significance comes from the fact that many examples in algebraic geometry are of this form, such as the
Segre embedding In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre.
Definition
The Segre map may be defined as the map ...
of a product of two
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s.
Definition
Given ''m'' and ''n'' and ''r'' < min(''m'', ''n''), the determinantal variety ''Y''
''r'' is the set of all ''m'' × ''n'' matrices (over a field ''k'') with rank ≤ ''r''. This is naturally an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
as the condition that a matrix have rank ≤ ''r'' is given by the vanishing of all of its (''r'' + 1) × (''r'' + 1)
minors. Considering the generic ''m'' × ''n'' matrix whose entries are
algebraically independent
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K.
In particular, a one element set \ is algebraically ind ...
variables ''x''
''i'',''j'', these minors are polynomials of degree ''r'' + 1. The ideal of ''k''
''i'',''j''">'x'' ''i'',''j''generated by these polynomials is a determinantal ideal. Since the equations defining minors are homogeneous, one can consider ''Y''
''r'' either as an
affine variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
in ''mn''-dimensional
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, or as a
projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
in (''mn'' − 1)-dimensional
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
.
Properties
The
radical ideal
In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is called ' ...
defining the determinantal variety is generated by the (''r'' + 1) × (''r'' + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10).
Assuming that we consider ''Y''
''r'' as an
affine variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
, its dimension is ''r''(''m'' + ''n'' − ''r''). One way to see this is as follows: form the product space
over
where
is the
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
of ''r''-planes in an ''m''-dimensional vector space, and consider the subspace
, which is a
desingularization
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characterist ...
of
(over the open set of matrices with rank exactly ''r'', this map is an isomorphism), and
is a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
over
which is isomorphic to
where
is the tautological bundle over the Grassmannian. So
since they are
birationally equivalent
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
, and
since the fiber of
has dimension ''nr''.
The above shows that the matrices of rank <''r'' contains the
singular locus of
, and in fact one has equality. This fact can be verified using that the radical ideal is given by the minors along with the
Jacobian criterion
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
*Jacobian matrix and determinant
* Jacobian elliptic functions
*Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler ...
for nonsingularity.
The variety ''Y''
''r'' naturally has an action of
, a product of
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
s. The problem of determining the
syzygies of
, when the
characteristic of the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
is zero, was solved by
Alain Lascoux
Alain Lascoux (17 October 1944 – 20 October 2013) was a French mathematician at the University of Marne la Vallée and Nankai University. His research was primarily in algebraic combinatorics, particularly Hecke algebras and Young tableaux.
L ...
, using the natural action of ''G''.
Related topics
One can "globalize" the notion of determinantal varieties by considering the space of linear maps between two vector bundles on an algebraic variety. Then the determinantal varieties fall into the general study of
degeneracy loci. An expression for the cohomology class of these degeneracy loci is given by the
Thom-Porteous formula, see (Fulton-Pragacz).
References
*
*
*
*
*{{cite book , first=Jerzy , last=Weyman , title=Cohomology of Vector Bundles and Syzygies , url=https://books.google.com/books?id=t_jdqfMMtnYC , date=2003 , publisher=Cambridge University Press , isbn=978-0-521-62197-7 , series=Cambridge Tracts in Mathematics , volume=149
Algebraic geometry
Algebraic varieties