Algebraic Geometry A First Course (3).pdf

LECTURE 12
More putations
Example . Determinantal Varieties
As in Lecture 9, let M be the projective space of nonzero m x n matrices up to
scalars and Mk c M the variety of matrices of rank k or less. To find the dimension
of M,, we introduce another incidence correspondence. We define
‘3’ c M x G(n - k, n)
Y = ((A, A): A c Ker(A))
(here, of course, we are viewing an m x n matrix A as a linear map K” -+ K”). The
point is that if we fix A the space of maps A: K”  -+ Km such that A c Ker(A) is just
the space Hom(K”/A, Km), so that fibers of the projection x,: Y + G(n - k, n) are
just projective spaces of dimension km - 1. We conclude that the variety ‘­I’ is
irreducible of dimension dim(G(n - k, n)) + km - 1 = k(m f n - k) - 1; since the
map nl: Y -+ M is generically one to one onto Mk, we deduce that the same is true
of Mk. One way to remember this is as the following proposition.
Proposition . The variety Mk c M of m x n matrices of rank at most k is
irreducible of codimension (m - k)(n - k) in M.
Exercise . Show that equality holds in Exercise .
Exercise . Use an analogous construction to estimate the dimension of the
spaces of(i) symmetric n x n matrices of rank at most k and (ii) skew-symmetric
n x n matrices of rank at most 2k.
152 12. More putations
Example . Fano Varieties
In Lecture 6 we described the Fano variety &,(X) parametrizing k-planes lying on
a variety X c P”. Here we will estimate its dimension, at least in the case of X a
general hypersurface of degree d in P”.
We do this by considering the space PN parametrizing all hypersurfaces of
n+d
d
degree in P”  (N here is d - 1, though as we shall see that won’t appear in
( )
putation), and setting up an incidence correspondence between hyper-
surfaces and planes. Specifically, we define the variety CDc PN x G(k, n) to be
0 = {(X, A): A c X}
so that the fiber of @ over a point X E P”  is just the Fano