RULED SURFACES

An irreducible projective variety of dimension 2 is called a projective surface. We call a surface S ⊂ ℙ^{n}
ruled if it is covered by a family of lines. Recall that the Grassmanian 𝔾(1,n) is a projective variety
parametrising lines in ℙ^{n}; there is a sub-variety I ⊂ ℙ^{n} × 𝔾(1,n) such that the fibre over any point of
𝔾(1,n) is a line in ℙ^{n}. We thus define a ruled surface by saying that there is a curve C ⊂ G(1,n) such that
S is the image p_{1}(p_{2}^{-1}(C) ∩ I).

We have already seen two examples in the previous lecture. The first (trivial) example is the cone. Let C
be a curve in ℙ^{n-1} and consider ℙ^{n-1} as obtained from ℙ^{n} by projecting from a point P in ℙ^{n}. Let S be the
closure of the inverse image of C under the morphism ℙ^{n} -{P}→ ℙ^{n-1}. Clearly, S is a ruled surface. A
slightly more interesting example is the surface Q in ℙ^{3} that is defined by WY - XZ = 0. As we
saw in the previous lecture, this is isomorphic to ℙ^{1} × ℙ^{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} ∩ Q is the union of the parametric lines (here we assume that
w0)

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 𝔾(2,n) for
various n, but the latter approach is additionally complicated by the slightly more complicated
geometry of 𝔾(2,n) as compared with the geometry of ℙ^{n}. Hence, we study ruled surfaces more
directly.

Consider a point A on a ruled surface S ⊂ ℙ^{n} and the projection p_{A} : ℙ^{n} -{A}→ ℙ^{n-1}. If L is any line in
S that does not contain A, then the image of L is again a line in ℙ^{n-1}. Thus the closure T of the image of
S -{A} is again a ruled surface providing that the image is a surface; the latter condition only fails when S
is the cone over a curve in ℙ^{n-1} as constructed above. When T is a surface it is 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 _{A} ⊂ ℙ^{n} × ℙ^{n-1}. The
morphism _{A} → ℙ^{n-1} identifies ℙ^{n-1} as the space of lines in ℙ^{n} that contain A or equivalently the space of
tangent directions to ℙ^{n} at A. As is obvious, any point B of ℙ^{n} such that BA determines a
unique such tangent direction (given by the line joining A and B) so that morphism → ℙ^{n}
is an isomorphism outside A. On the other hand the fibre of this morphism over A is all of
ℙ^{n-1}.

If X is any irreducible variety in ℙ^{n} that properly contains A, then let _{A} ⊂_{A} denote the closure of
the graph of the restricted morphism p_{A}|_{X} : X -{A}→ ℙ^{n-1}. When X is smooth at A the fibre of
_{A} → 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 define what one can think of as the locus of lines tangent to X at A; but this is
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 _{A} → X was called the 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 _{A} → X is called the blow-up of X at
A.

Returning to the ruled surface S, let _{A} → 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 _{A} is a smooth curve
E_{A}ℙ^{1} which sits in ℙ^{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 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 ℙ^{1}.

Let C_{d} denote the rational normal curve of degree d in ℙ^{d}. Fix an isomorphism f_{d} : ℙ^{1} → C_{d}. The
projective spaces ℙ^{d} and ℙ^{e} can be put as disjoint linear spaces in the projective space ℙ^{d+e+1}; let us denote
these by L^{d} and M^{e} respectively. Each point A of ℙ^{d+e+1} which is not contained in L^{d} ∩ 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
ℙ^{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 ℙ^{1} gives a ruled surface
S_{d,e} in ℙ^{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
ℙ^{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 ∩ S_{d,e} \ 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 ∩ S_{d,e} in general consists of C_{d} and e lines L_{1}, …, 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 ℙ^{n} will have a linear span of
dimension 2e - 1 as long as n ≥ 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 ℙ^{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 ℙ^{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 ℙ^{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 ⊂ ℙ^{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 ℙ^{n} defined by the vanishing of F. One
can show that V (F) ∩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 ⋅D); where (C ⋅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 = ℙ^{n-1} ∩ S is the intersection of S with a hyperplane in ℙ^{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 ⊂ ℙ^{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.