OTHER SURFACES

An irreducible projective variety S of dimension 2 is called a projective surface.

One way to study surfaces is to think of them as “moving curves” with one parameter. More precisely, we can ask for a curve C and an irreducible closed subvariety (which we can think of as a correspondence) T ⊂ C ×S such that the fibre over each point of C is a curve in S. We have already seen one such example, that of ruled surfaces in the previous section. Going on from there it is natural to take up the study of surfaces where a C and T can be found where the (general) fibre is a curve of genus 1.

A different way to approach the study of surfaces is to extend our knowledge of plane curves and study
surfaces in ℙ^{3} or equivalently sub-varieties of ℙ^{3} defined as the vanishing locus of a single irreducible
homogeneous polynomial F(W,X,Y,Z).

One may also try to generalise the Riemann-Hürwitz approach by thinking of surfaces given in the form
of a finite morphism S → ℙ^{2}. In the case of curves given as C → ℙ^{1} the topology is rather simply described
as the branch locus is a finite set of points in ℙ^{1}. However, the situation of surfaces is made more
complicated by the numerous types of curves in ℙ^{2}. As a result this approach is much more
difficult.

Let us thus begin with surfaces in ℙ^{3}. We have already studied the surfaces in ℙ^{3} defined by equations of
degree 2, so let us continue with surfaces of degree 3. We will show that a smooth cubic surface S in ℙ^{3} has
27 lines which are in a rather special position with respect to each other and contain a configuration known
as the “Schlafly double-six”. Before we do that let us take the projection of S from a general line M in ℙ^{3}.
We obtain a morphism S \ S ∩ M → ℙ^{1} where S ∩ M consists of 3 distinct points. Let be the
closure of the graph of this morphism. As seen earlier the fibre E_{i} of → S over a point i in
S ∩ M consists of a copy of ℙ^{1} which can be identified with the projective space of directions
tangent to S at i. Moreover, each of these copies is mapped isomorphically under the natural map
π : → ℙ^{1}. Each fibre of π is a plane cubic curve; we have seen that this is a curve of genus 1.
Thus the study of the cubic surface generalises the study of ruled surfaces as well. An easy
exercise in computing the degree of the dual variety shows that if M is a general line (and the
characteristic is zero), then there are exactly 6 singular fibres and each such fibre is a nodal
plane cubic. The following study of the cubic surface can therefore be interpreted in terms of
the group structure on the generic fibre of π; those interested may try to follow this approach
later.

We first show that there is one cubic surface which has exactly 27 lines. To begin with let us choose a
smooth cubic surface S that contains a line L; for example we can take the surface defined by the equation
WX(W - X) = Y Z(Y - Z) which contains the line defined by W = Y = 0. More generally, the surface
S_{0} and contains the lines L_{a,b} which are defined by the equations W - aX = 0 Y - bZ = 0
where (a,b) run over the set {0,1,∞}^{2}. The projection of S_{0} from the line L_{∞,∞} is given by the
morphism

which clearly extends to a morphism f : S_{0} → ℙ^{1}. The fibre of f over the point (t : 1) of ℙ^{1} is the conic C_{t}
defined by Y (Y -Z) = tW(W -tZ) in the plane defined by X = tZ. This conic becomes a pair of straight
lines for t in the set {0,∞,1,ω,ω^{2}} where ω is a primitive cube root of 1; this gives all the
lines that are contained in the fibres of f or equivalently all lines that meet L_{∞,∞}. We have

The following combinatorial description helps us to identify this as a “double-six”. Consider the 6-tuples

A double-six consists of a pair of ordered 6-tuples of lines (P_{0},…,P_{6}) and (Q_{0},…,Q_{6})
such that each P_{i} meets all the Q_{j} except Q_{i}.

Now, for each i and j Consider the intersection of the plane spanned by P_{i} and Q_{j} with that spanned by P_{j}
and Q_{i}; this gives a line R_{ij} which meets all the four lines. In the previous terminology we have
R_{0i} = N_{i} and R_{lm} = M_{ijk} where {i,j,k,l,m} = {1,2,3,4,5}. This gives us 6 + 6 + = 27
lines.

The projective space of dimension - 1 = 19 parametrises all homogeneous cubic polynomials in 4
variables or equivalently cubic surfaces in ℙ^{3}. As we have seen before there is a 4 dimensional variety
𝔾(1, ℙ^{3}) that parametrises lines in ℙ^{3}. Let I denote the sub-variety of ℙ^{19} × 𝔾(1, ℙ^{3}) that consists of pairs
(S,L) such that L is contained in S. For each L we obtain 4 linear conditions which are the equations that
define the locus of cubics that contain L. In other words the fibre of I over each point of 𝔾(1, ℙ^{3}) is a linear
ℙ^{15} in ℙ^{19}; so I is also 19-dimensional. We have shown that the fibre of I → ℙ^{19} consists of finitely many
points over the point corresponding to the surface S_{0}. It follows that this is a surjective morphism. In
particular, each cubic surface contains a line. One can then duplicate the argument given above
to show that the configuration of lines in the cubic form is generated from the “double-six”
as described above. (Warning: There could be some additional incidence conditions in some
cases. When three lines among the 27 lie in a plane then they may form a “star” instead of a
“triangle”.)

Perhaps the first Indian mathematician in recent times who developed expertise in the study of algebraic surfaces was C. P. Ramanujam of the TIFR. He posed some very interesting questions which have led to a lot of research by R. V. Gurjar, A. R. Shastri and C. R. Pradeep. Their work involves the study of surfaces which occur as genus 1 fibres over curves; this was initiated by K. Kodaira. Such surfaces have also been studied M. M. Nori of Mumbai University. The study of singular surfaces (or surface singularities) is where V. Srinivas of TIFR has made significant contributions. Kodaira had classified surfaces according to their invariants much as curves are classified in terms of the genus. The extension of this work (to varieties of higher dimension) by Mori, Kawamata, Kollár, Shokurov and others was an important development in the late ’80s and early ’90s. Unfortunately, no Indian mathematician played a significant role in this development of ideas.