MORPHISMS AND CORRESPONDENCES

A correspondence between a pair of algebraic varieties X and Y can loosely be defined as a
closed sub-variety Z of the product X × Y . One obvious example is the graph Γ_{f} of a morphism
f : X → Y . However, note that there is a natural symmetry which interchanges correspondences
between X and Y with those between Y and X; the notion of morphism does not have such a
symmetry.

The problem we shall consider today is how one can construct interesting correspondences. So perhaps an
easier question could be how does one construct interesting morphisms. Since we are only considering
quasi-projective varieties, a morphism X → Y can be thought of as a morphism X → ℙ^{n} which happens to
land inside the sub-variety Y ⊂ ℙ^{n}. We have thus simplified the question; we now only ask how one
constructs morphisms from X to ℙ^{n}.

Since X is itself a sub-variety of ℙ^{m} for some m, we can try to construct morphisms from ℙ^{m} → ℙ^{n} and
go on from there. A fundamental example of such a morphism is the Veronese embedding ℙ^{m} → ℙ^{-1}
given by

Another very important morphism is the projection ℙ^{m} \ ℙ^{k} → ℙ^{m-k-1} given by

This morphism is cannot be extended along the sub-variety Lℙ^{k} ⊂ ℙ^{m} given by the equations
T_{0} = = T_{m-k-1} = 0. In case our original variety X does not intersect L, then the morphism is defined
on X. With an eye on the notion of correspondences we can consider the closure in ℙ^{m} × ℙ^{m-k-1}
of the graph B of the above morphism. We note that B → ℙ^{m} is an isomorphism over the
open sub-variety ℙ^{m} \ L. The collection of all points in ℙ^{m} that are associated with a given
(x_{0} : : x_{m-k-1}) in ℙ^{m-k-1} forms the linear subspace M_{(x0::xm-k-1)}ℙ^{k+1} given by the
equations x_{j}T_{i} = x_{i}T_{j}. It is clear that L ⊂ M_{(x0::xm-k-1)}. In other words, we can think of B as
the space of all pairs (p,M) where M is a linear subspace of dimension k + 1 in ℙ^{m} such that
M contains L and the point p lies on M; when p is not in L, the linear span of p and L is
M.

More generally, we can perform a “blow-up” of any X such that no component of X is contained in L.
Consider the restriction of the projection morphism to X \L. We obtain a space B_{X} ⊂ X × ℙ^{m-k-1} which
is the closure of the graph of this morphism; this is called the blow-up of X along X ∩ L. The resulting
correspondence is called a rational map on X.

The projection B_{X} → X is an isomorphism over the open sub-variety X \ (X ∩L). However, it could be
an isomorphism on a bigger open sub-variety as well. In certain situations the first projection B_{X} → X is an
isomorphism and then we obtain a morphism X → ℙ^{m-k-1}. It is an intriguing exercise to show that every
morphism X → Y from a quasi-projective variety to another can be obtained by these means. In
other words, given X ⊂ ℙ^{m} and Y ⊂ ℙ^{n} and a morphism f : X → Y , there is a d and a linear
subspace L in ℙ^{-1} such that the morphism f is obtained as the composition of the Veronese
embedding X → ℙ^{-1} with the projection from L, which is so chosen that B_{
X} → X is an
isomorphism.

An important example is the case when X is a smooth curve; by this we mean a smooth
irreducible projective variety of dimension 1. Let p be a point of X and consider the projection
from p. One can see that B_{X} is isomorphic to X; since there is “only one point missing” from
X - p, the closure of this variety can only yield one more point! (Note the contrast between the
algebraic context and the continuous context.) As an example, let us consider the curve in ℙ^{3}
defined by the equations X^{2} + Y ^{2} = Z^{2} and aX^{2} + bY ^{2} = W^{2}. Projecting this form a point on it
gives a plane cubic curve in ℙ^{3}. As an exercise you could try to write the equation of such a
cubic.

Consider the variety B_{ℙm} in ℙ^{m} × ℙ^{m-k-1} which is obtained as the graph of the rational map (projection)
ℙ^{m} \ ℙ^{k} → ℙ^{m-k-1}. It is also useful to think of it as a correspondence from ℙ^{m-k-1} to ℙ^{m}. As seen above it
can be thought of as the family of all k + 1 dimensional linear spaces in ℙ^{m} that contain the fixed ℙ^{k} from
which we are projecting. This naturally leads us the the question of whether we can find the family of all
ℙ^{k}’s in ℙ^{m}.

For example, let us consider the sub-variety I of ℙ^{n} × ℙ^{n} defined by the equation ∑
X_{i}Y _{i} = 0, where
(X_{0} : : X_{n}) and (Y _{0} : : Y _{n}) are the projective co-ordinates on the two ℙ^{n}’s. For each point
(a_{0} : : a_{n}) of ℙ^{n}, the equation ∑
a_{i}Y _{i} = 0 defines a linear sub-space of dimension n - 1 in ℙ^{n}.
Conversely, any such linear space is defined by such an equation. Hence, we see that I can be thought of the
variety parametrising such linear subspaces of dimension n - 1 — indeed, by symmetry this
is true whichever way we look at this correspondence; with respect to the first factor or the
second.

To take up a more asymmetric situation, let us consider the ℙ^{-1} whose projective co-ordinates we
think of as tuples (X_{ij}) with the relations X_{ii} = 0 and X_{ij} + X_{ji} = 0. We then define the sub-variety I of
ℙ^{-1} × ℙ^{n} by the equations X_{
ij}Y _{k} + X_{jk}Y _{i} + X_{ki}Y _{j} = 0 as i, j, k run over all possible values from 0 to
n.

For each point (b_{0} : : b_{n}) of ℙ^{n}, the linear equations on the X_{ij} given by substituting b_{i} in place of Y _{i}
give a linear subspace of ℙ^{n}; it is an exercise to show that this is a linear space of dimension n - 1. (Hint:
Try to show that if (x_{ij}) is a tuple satisfying these equations then x_{ij} = b_{i}c_{j} - b_{j}c_{i} for some c_{i}). The
dimension of I is thus 2n- 1. One can similarly show that the linear space in ℙ^{n} defined by the equations
obtained on substituting X_{ij} by any tuple a_{ij} (such that a_{ii} = 0 and a_{ij} + a_{ji} = 0) is of dimension 1 if it is
non-empty.

It follows that the image of I in ℙ^{-1} is a variety of dimension 2n - 2; this is the Grassmannian
variety 𝔾(1,n). Plücker did the necessary elimination theory to write down the explicit equations for this
variety and so can you! One can generalise this approach to construct the Grassmannian variety 𝔾(k,n)
which parametrises ℙ^{k}’s in ℙ^{n}. The study of the geometry of the Grassmannian varieties (and other similar
varieties) has attracted numerous algebraic geometers including C. S. Seshadri, C. Musili, V. Lakshmibai,
A. Ramanathan, K. N. Raghavan and P. Sankaran.

More generally, suppose one is trying to parametrise all sub-varieties of “type” X in projective space ℙ^{n}.
If the homogeneous co-ordinate ring of X is R_{X}, one can say that Y has the same type as X if
the graded components satisfy dim(R_{Y })_{m} = dim(R_{X})_{m} for all sufficiently large m. One can
find a degree d so that X is defined by equations of degree d. Consider the d-tuple Veronese
embedding of ℙ^{n} in ℙ^{-1}; denote the image by P_{
d}. The sub-variety X in P_{d} is defined by linear
equations and is thus given by the intersection of P_{d} with a suitable linear space ℙ^{k} in the
ambient projective space. It is natural to expect (though it requires some effort to prove) that
other sub-varieties in ℙ^{n} of the same type as X are also obtained by intersecting P_{d} with linear
subspaces of the same dimension; in other words, every Y with the property given above is also
defined by same number of equations of degree d. However, it is unlikely (give an example!)
that all linear spaces of dimension k will intersect P_{d} in a variety of type X. The condition
that fixes the type will be a closed condition in the sense that there is a closed sub-variety of
𝔾(k,- 1) for which the type will be preserved. This variety parametrises sub-varieties in P_{d}ℙ^{n}
of the same type as X and is called the Hilbert scheme of subvarieties of type X in ℙ^{n}. One
can similarly construct Hilbert schemes of sub-varieties of a general projective variety Y . The
natural incidence between the Hilbert scheme and Y gives an example of a very interesting
correspondence.

To conclude this discussion let us take a simple example of the Hilbert scheme—the space of conics in ℙ^{2}.
Consider the sub-variety of ℙ^{2} × ℙ^{5} given by the equation

This exhibits ℙ^{5} as the space of conics in ℙ^{2}.