Algebraic Geometry A First Course (2).pdf

LECTURE 9
Determinantal Varieties
In this lecture we will introduce a large and important class of varieties, those
whose equations take the form of the minors of a matrix. We will see that many of
the varieties we have looked at so far-Veronese varieties, Segre varieties, rational
normal scrolls, for example-are determinantal.
Generic Determinantal Varieties
We will start with what is called the generic determinantal variety. Let M be
the projective space Pm”’ associated to the vector space of m x n matrices. For
each k, we let Mk c M be the subset of matrices of rank k or less; since this is just
mon zero locus of the (k + 1) x (k + 1) minor determinants, which are
homogeneous polynomials of degree k + 1 on the projective space M, Mk is a
projective variety.
Note. More or less by definition, the (k + 1) x (k + 1) minors cut out the variety
Mk set theoretically. The stronger statement-that they generate the homogeneous
ideal Z(M,) of M,-is also true, but is nontrivial to prove.
Example . Segre Varieties
The simplest example of a generic determinantal variety, the case k = 1, is a variety
we have already encountered. The basic observation here is that an m x n matrix
(Zi, j) will be of rank 1 if and only if it is expressible as a product 2 = ‘U . W, where
u = (Ui,..., U,,,) and W = (W;, . . . . W,) are vectors. We see from this that the
subvariety M, c M = P’ mn-l is just the Segre variety, that is, the image of the Segre
Linear Determinantal Varieties in General 99
map
pm-1
a: x p-1 ~ pmn-1.
(In intrinsic terms, a homomorphism A: K” + K” of rank 1 is determined up
to scalars by its kernel Ker(A) E (P-i)* and its image Im(A) E P-l.) Thus, we can
represent the quadric surface C,,, = a(P’ x P’) c P3 as
{cz1:1~;:I=o)
x1,, =
and similarly the Segre threefold C,, 1 = o(P’ x P’) c Ps may be realized as
of
Example . Secant Varieties Segre Varieties
We can in fact describe the other gen