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\begin{document}

\title{Lecture VII\\ Ruled Surfaces}

\author{Kapil Hari Paranjape}
\maketitle

An irreducible projective variety of dimension 2 is called a projective
surface. We call a surface $S\subset\bbP^n$ ruled if it is covered by
a family of lines. Recall that the Grassmanian $\bbG(1,n)$ is a
projective variety parametrising lines in $\bbP^n$; there is a sub-variety
$I\subset\bbP^n\times\bbG(1,n)$ such that the fibre over any point of
$\bbG(1,n)$ is a line in $\bbP^n$. We thus define a ruled surface by
saying that there is a curve $C\subset G(1,n)$ such that $S$ is the image
$p_1(p_2^{-1}(C)\cap I)$.

We have already seen two examples in the previous lecture. The first
(trivial) example is the cone. Let $C$ be a curve in $\bbP^{n-1}$ and
consider $\bbP^{n-1}$ as obtained from $\bbP^n$ by projecting from a
point $P$ in $\bbP^n$. Let $S$ be the closure of the inverse image of
$C$ under the morphism $\bbP^n-\{P\}\to\bbP^{n-1}$. Clearly, $S$ is a
ruled surface. A slightly more interesting example is the surface $Q$
in $\bbP^3$ that is defined by $WY-XZ=0$. As we saw in the previous
lecture, this is isomorphic to $\bbP^1\times\bbP^1$. In fact, if
$A=(w:x:y:z)$ is any point on $Q$, then the tangent plane $T_A$ to $Q$
at $A$ is defined by $yW-zX+wY-xZ=0$. One computes that the
intersection $T_A\cap Q$ is the union of the parametric lines (here
we assume that $w\neq 0$)
\[
	t \mapsto (w:x:xt:wt) \text{~and~}
	t \mapsto (w:wt:zt:z)
\]
It follows that $Q$ is a ruled surface in two different ways.

Now one could argue that the study of ruled surfaces is identical to
the study of curves in $\bbG(2,n)$ for various $n$, but the latter
approach is additionally complicated by the slightly more complicated
geometry of $\bbG(2,n)$ as compared with the geometry of $\bbP^n$.
Hence, we study ruled surfaces more directly.

Consider a point $A$ on a ruled surface $S\subset\bbP^n$ and the
projection $p_A:\bbP^n-\{A\}\to\bbP^{n-1}$. If $L$ is any line in $S$
that does not contain $A$, then the image of $L$ is again a line in
$\bbP^{n-1}$. Thus the closure $T$ of the image of $S-\{A\}$ is again a
ruled surface providing that the image \emph{is} a surface; the
latter condition only fails when $S$ is the cone over a curve in
$\bbP^{n-1}$ as constructed above. When $T$ is a surface it is
\emph{not} isomorphic to $S$; when $A$ is a smooth point of $S$,
the correspondence between $T$ and $S$ is called an elementary
transformation from $T$ to $S$. 

Recall that the closure of the graph of $p_A$ is an $n$-dimensional
sub-variety $\tilde{\bbP^n}_A\subset\bbP^n\times\bbP^{n-1}$. The
morphism $\tilde{\bbP^n}_A\to\bbP^{n-1}$ identifies $\bbP^{n-1}$ as the
space of lines in $\bbP^n$ that contain $A$ or equivalently the space
of tangent directions to $\bbP^n$ at $A$. As is obvious, any point
$B$ of $\bbP^n$ such that $B\neq A$ determines a unique such tangent
direction (given by the line joining $A$ and $B$) so that morphism
$\tilde{\bbP^{n}}\to\bbP^n$ is an isomorphism outside $A$. On the
other hand the fibre of this morphism over $A$ is all of
$\bbP^{n-1}$.

If $X$ is any irreducible variety in $\bbP^n$ that properly contains $A$,
then let $\tilde{X}_A\subset\tilde{\bbP^n}_A$ denote the closure of the
graph of the restricted morphism $p_A|_X:X-\{A\}\to\bbP^{n-1}$. When
$X$ is smooth at $A$ the fibre of $\tilde{X}_A\to X$ over the point $A$
can be realised as the space of tangent directions to $X$ at $A$. When
$X$ is singular, this could be used to \emph{define} what one can think of
as the locus of lines tangent to $X$ at $A$; but this is \emph{not}
the collection of all lines in the Zariski tangent space of $X$ at
$A$ when $X$ is singular at $A$. In order to avoid confusion, the
fibre of $\tilde{X}_A\to X$ was called the \emph{apparent locus} of $X$ at
$A$ in classical geometry. Its algebraic description is as the ``proj''
of the associated graded ring obtained from the local ring of $X$ at $A$.
The morphism $\tilde{X}_A\to X$ is called the blow-up of $X$ at $A$.

Returning to the ruled surface $S$, let $\tilde{S}_A\to S$ be the
blow-up morphism as above. Let us further assume that $A$ is a smooth
point of $S$. By the above description, the inverse image of $A$ in
$\tilde{S}_A$ is a smooth curve $E_A\cong\bbP^1$ which sits in
$\bbP^{n-1}$ as the locus of all directions which are tangent to $S$.
Now, if $L$ is a line in $S$ which contains $A$ then $L-\{A\}$ is
collapsed to a point $B\in E_A$ under $p_A$. In general, there is a unique such
line (in fact $Q$ is essentially the only case where this is not
the case). Thus, the elementary transformation from $S$ to $T$ with
centre $A$ is obtained by replacing the line $L_A$ which contains $A$
with a point $B$ which lies on a line $E_A$ which takes the place of
$L_A$ in the ruling of $T$. 

Let us temporarily specialise to the case when the base of the ruling
is isomorphic to $\bbP^1$.

Let $C_d$ denote the rational normal curve of degree $d$ in $\bbP^d$.
Fix an isomorphism $f_d:\bbP^1\to C_d$. The projective spaces $\bbP^d$
and $\bbP^e$ can be put as disjoint linear spaces in the projective
space $\bbP^{d+e+1}$; let us denote these by $L^d$ and $M^e$ respectively.
Each point $A$ of $\bbP^{d+e+1}$ which is not contained in $L^d\cap
M^e$ gives rise (uniquely) to two points $B$ and $C$ in these linear
spaces such that $A$ lies on the line joining $B$ and $C$ in
$\bbP^{d+e+1}$. The linear span of $M^e$ and $A$ meets $L^d$ in a
unique point $B$; similarly for $C$. The intersection of the span of
$M^e$ and $A$ with the span of $L^d$ and $A$ is the line joining $B$
and $C$. In particular, we see that the locus of lines joining
$f_d(t)$ with $f_e(t)$ as $t$ varies over $\bbP^1$ gives a ruled
surface $S_{d,e}$ in $\bbP^{d+e-1}$. A general hyperplane meets each
of these lines in one point. It follows that the intersection of a
general hyperplane with $S_{d,e}$ gives a rational normal curve of degree
$d+e$ in $\bbP^{d+e}$. Consider a hyperplane $H$ that contains $L^d$.
Its intersection with $S_{d,e}$ already contains $C_d$. Let $A$ be a
point of $H\cap S_{d,e}\setminus C_d$. Let $L$ be the line in
$S_{d,e}$ that contains $A$. Since $L$ meets $H$ and $A$ as well as
the endpoint of $L$ that lies in $C_d$, we see that $L$ lies in $H$.
Thus (from degree considerations) it follows that $H\cap S_{d,e}$ in
general consists of $C_d$ and $e$ lines $L_1$, \dots, $L_e$ from the
ruling. Similarly, a general hyperplane containing $M^e$ intersects
$S_{d,e}$ in $C_e$ and $d$ lines from the ruling on $S_{d,e}$. Now,
$e$ general lines in a projective space $\bbP^n$ will have a linear
span of dimension $2e-1$ as long as $n\geq 2e-1$. Thus, if $d>e$, the
span of any $e$ lines in the ruling of $S_{d,e}$ determines a linear
subspace of dimension $2e-1$ in $\bbP^{d+e+1}$; in particular, there
are many hyperplanes that contain such an $e$-tuple of lines. The
residual intersection of such a hyperplane with $S_{d,e}$ gives a
curve of degree $d$. One can easily show that this is another
rational normal curve of degree $d$ which too spans a linear subspace
of $\bbP^{d+e+1}$ which is disjoint from $M^e$. On the other hand
we cannot take the linear span of $d$ general lines in this situation
since their span will be all of $\bbP^{d+e+1}$. Using this one can
show that $C_e$ is the curve of least degree in $S_{d,e}$. Similarly,
one can show that the elementary transform of $S_{d,e}$ from a point
not on $C_e$ results in $S_{d-1,e}$, whereas the elementary transform
of $S_{d,e}$ from a point on $C_e$ results in $S_{d,e-1}$.

In order to explain the next statement we need to introduce the
notion of self-intersection of a curve on a surface $S$. Let
$S\subset\bbP^n$ be a surface and $C$ a curve on $S$. Let $F$ be a
general homogeneous polynomial of large degree that vanishes on $C$
and let $V(F)$ denote the hypersurface in $\bbP^n$ defined by the
vanishing of $F$. One can show that $V(F)\cap S$ is of the form $C+D$
for $D$ an irreducible curve on $S$. We define the self-intersection
$(C.C)$ of $C$ as $\deg(C)\deg(F)-(C\cdot D)$; where $(C\cdot D)$
denotes the intersection of $C$ and $D$ as defined earlier. As in the
case of most such definitions the main work goes into proving that
this number is independent of the choice of $F$; in fact it is also
independent of the embedding of $S$ in projective space. We note that if
$H=\bbP^{n-1}\cap S$ is the intersection of $S$ with a hyperplane
in $\bbP^n$ then $(H.H)=\deg(H)=\deg(S)$.

Returning to the ruled surfaces $S_{e,d}$, let $L$ be a line which is
the ruling of $S_{e,d}$. A general hyperplane that contains $L$ will
meet all other lines in the ruling transversely at one point. It
follows that the intersection of such a hyperplane with $S$ is of the
form $C+L$ where $C$ meets $L$ in exactly one point. Thus $(L.L)=0$.
Now some easy calculations show that $(C_e.C_e)=e-d$ in $S_{d,e}$. In
particular, when $d>e$ this is a negative number. This is not
un-correlated with what we showed above---viz.\ the curve $C_e$ does
not ``move'' in this case.

Let us move on to more general ruled surfaces. Let $S\subset\bbP^n$ be
a ruled surface. We wish to study curves $H$ on $S$ with the property
that they meet each line in the ruling of $S$ transversely; such
curves are called ``sections'' of the ruling for obvious reasons.
By a procedure similar to the one used to construct $S_{d,e}$, it is
not difficult to construct surfaces which have sections $H$ where
$(H.H)$ can become as small as one wants. In the interests of
restricting one's attention to ``interesting'' surfaces one can look
at the case where $(H.H)$ is bounded below. One such restriction that
has proved very worthwhile is to require that $(H.H)>0$ for every
section $H$. Such ruled surfaces are called ``stable''. A slight weakening
of this is to allow equality as well; in that case the ruled surface is
called ``semi-stable''.

M. S. Narasimhan and C. S. Seshadri were able to prove a remarkable
topological description of stable (and semi-stable) ruled surfaces.
This description could be used to find a parametrisation of all such
surfaces. The study of these parameter spaces (or moduli spaces) has
been the subject of numerous papers by algebraic geometers at the
TIFR such as S. Ramanan, A. Ramanathan, V. B. Mehta, N. Nitsure, Y.
Holla and others.

\end{document}