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\begin{document}
\title{Lecture I\\ Morphisms and Correspondences}
\author{Kapil Hari Paranjape}
\maketitle
A correspondence between a pair of algebraic varieties $X$ and $Y$ can
loosely be defined as a closed sub-variety $Z$ of the product $X\times Y$.
One obvious example is the graph $\Gamma_f$ of a morphism $f:X\to Y$.
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 \emph{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 $X\to Y$ can be
thought of as a morphism $X\to \bbP^n$ which happens to land inside
the sub-variety $Y\subset\bbP^n$. We have thus simplified the
question; we now only ask how one constructs morphisms from $X$ to
$\bbP^n$.
Since $X$ is itself a sub-variety of $\bbP^m$ for some $m$, we can
try to construct morphisms from $\bbP^m\to \bbP^n$ and go on from there.
A fundamental example of such a morphism is the Veronese embedding
$\bbP^m\to \bbP^{\binom{m+d}{d}-1}$ given by
\[
(X_0:\dots:X_m) \mapsto
(X_0^d:\dots:\text{monomials of degree d}:\dots:X_m^d)
\]
Another very important morphism is the projection
$\bbP^m\setminus\bbP^k\to\bbP^{m-k-1}$ given by
\[
(T_0:\dots:T_m) \mapsto (T_0:\dots:T_{m-k-1})
\]
This morphism is cannot be extended along the sub-variety
$L\cong\bbP^k\subset\bbP^m$ given by the equations $T_0=\dots=T_{m-k-1}=0$.
In case our original variety $X$ does not intersect
$L$, then the morphism \emph{is} defined on $X$.
With an eye on the notion of correspondences we can consider
the closure in $\bbP^m\times\bbP^{m-k-1}$ of the graph $B$ of the above
morphism. We note that $B\to\bbP^m$ is an isomorphism over the open
sub-variety $\bbP^m\setminus L$. The collection of all points in
$\bbP^m$ that are associated with a given $(x_0:\dots:x_{m-k-1})$ in
$\bbP^{m-k-1}$ forms the linear subspace $M_{(x_0:\dots:x_{m-k-1})}\cong\bbP^{k+1}$
given by the equations $x_j T_i=x_i T_j$. It is clear that $L\subset
M_{(x_0:\dots:x_{m-k-1})}$. 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 $\bbP^m$ 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 $X\setminus L$. We obtain a
space $B_X\subset X\times\bbP^{m-k-1}$ which is the closure of the graph
of this morphism; this is called the \emph{blow-up} of $X$ along
$X\cap L$. The resulting correspondence is called a
\emph{rational map} on $X$.
The projection $B_X\to X$ is an isomorphism over the open sub-variety
$X\setminus (X\cap L)$. However, it \emph{could} be an isomorphism
on a bigger open sub-variety as well. In certain situations the
first projection $B_X\to X$ is an isomorphism and then we obtain a
morphism $X\to \bbP^{m-k-1}$. It is an intriguing exercise to show
that \emph{every} morphism $X\to Y$ from a quasi-projective variety
to another can be obtained by these means. In other words, given
$X\subset\bbP^m$ and $Y\subset\bbP^n$ and a morphism $f:X\to Y$, there
is a $d$ and a linear subspace $L$ in $\bbP^{\binom{m+d}{m}-1}$ such
that the morphism $f$ is obtained as the composition of the Veronese
embedding $X\to\bbP^{\binom{m+d}{m}-1}$ with the projection from $L$,
which is so chosen that $B_X\to X$ 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 $\bbP^3$ 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 $\bbP^3$. As an exercise you could try to write the
equation of such a cubic.
Consider the variety $B_{\bbP^m}$ in $\bbP^m\times\bbP^{m-k-1}$
which is obtained as the graph of the rational map (projection)
$\bbP^m\setminus\bbP^k\to\bbP^{m-k-1}$. It is also useful to think of
it as a correspondence from $\bbP^{m-k-1}$ to $\bbP^m$. As seen above
it can be thought of as the family of all $k+1$ dimensional linear
spaces in $\bbP^m$ that contain the fixed $\bbP^k$ from which we are
projecting. This naturally leads us the the question of whether we can
find the family of all $\bbP^k$'s in $\bbP^m$.
For example, let us consider the sub-variety $I$ of $\bbP^n\times\bbP^n$
defined by the equation $\sum X_iY_i=0$, where $(X_0:\dots:X_n)$ and
$(Y_0:\dots:Y_n)$ are the projective co-ordinates on the two
$\bbP^n$'s. For each point $(a_0:\dots:a_n)$ of $\bbP^n$, the
equation $\sum a_i Y_i=0$ defines a linear sub-space of dimension
$n-1$ in $\bbP^n$. 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
$\bbP^{\binom{n+1}{2}-1}$ 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
$\bbP^{\binom{n+1}{2}-1}\times\bbP^{n}$ 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 $(b_0:\dots:b_n)$ of $\bbP^n$, the linear equations
on the $X_{ij}$ given by substituting $b_i$ in place of $Y_i$ give a
linear subspace of $\bbP^n$; 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 $\bbP^n$ 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 $\bbP^{\binom{n+1}{2}-1}$
is a variety of dimension $2n-2$; this is the Grassmannian variety
$\bbG(1,n)$. Pl\"ucker 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
$\bbG(k,n)$ which parametrises $\bbP^k$'s in $\bbP^n$. 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 $\bbP^n$. 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 $\bbP^n$ in $\bbP^{\binom{n+d}{d}-1}$;
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 $\bbP^k$ in the ambient projective space. It is
natural to expect (though it requires some effort to prove) that other
sub-varieties in $\bbP^n$ of the same type as $X$ are also obtained by
intersecting $P_d$ with linear subspaces of the same dimension; in other
words, \emph{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 \emph{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 $\bbG(k,\binom{n+d}{d}-1)$ for which the type will be preserved.
This variety parametrises sub-varieties in $P_d\cong\bbP^n$ of the
same type as $X$ and is called the Hilbert scheme of subvarieties of
type $X$ in $\bbP^n$. 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 $\bbP^2$. Consider the sub-variety of
$\bbP^2\times\bbP^5$ given by the equation
\[
Y_0 X_0^2+ Y_1 X_0X_1 + Y_2 X_1^2 + Y_3 X_2 X_0 + Y_4 X_2 X_1 +
Y_5 X_2^2 = 0
\]
This exhibits $\bbP^5$ as the space of conics in $\bbP^2$.
\end{document}