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\title{Lecture VIII\\ Other Surfaces}

\author{Kapil Hari Paranjape}
\maketitle

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) $T\subset C\times S$ 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 $\bbP^3$ or
equivalently sub-varieties of $\bbP^3$ defined as the vanishing locus of
a single irreducible homogeneous polynomial $F(W,X,Y,Z)$.

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

Let us thus begin with surfaces in $\bbP^3$. We have already studied
the surfaces in $\bbP^3$ 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 $\bbP^3$ 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 $\bbP^3$. We obtain a
morphism $S\setminus S\cap M \to\bbP^1$ where $S\cap M$ consists of 3
distinct points. Let $\tilde{S}$ be the closure of the graph of this
morphism. As seen earlier the fibre $E_i$ of $\tilde{S}\to S$ over a
point $i$ in $S\cap M$ consists of a copy of $\bbP^1$ 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 $\pi:\tilde{S}\to\bbP^1$. Each fibre of $\pi$ 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
$\pi$; those interested may try to follow this approach later.

We first show that there is \emph{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 $WX(W-X)=YZ(Y-Z)$ 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 $\{0,1,\infty\}^2$.
The projection of $S_0$ from the line $L_{\infty,\infty}$ is given by the morphism
\[ (W:X:Y:Z) \mapsto (X:Z) = (Y(Y-Z):W(W-X)) \]
which clearly extends to a morphism $f:S_0\to\bbP^1$. The fibre of
$f$ over the point $(t:1)$ of $\bbP^1$ 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
$\{0,\infty,1,\omega,\omega^2\}$ where $\omega$ 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 $L_{\infty,\infty}$. We have 
\begin{eqnarray*}
	C_0 & = & L_{\infty,0}\cup L_{\infty,1} \\
	C_{\infty} & = & L_{0,\infty} \cup L_{1,\infty} \\
	C_1 & = & L_{1,1}\cup V(X-Z,Y+W-Z) \\
	C_{\omega} & = &
	V(X-\omega Z,Y-\omega^2 W)\cup
	V(X-\omega Z,Y+\omega^2 W-Z)\\
	C_{\omega^2} & = &
	V(X-\omega^2 Z,Y-\omega W)\cup
	V(X-\omega^2 Z,Y+\omega W-Z)
\end{eqnarray*}
The line $L_{0,0}$ meets only the following 5 lines from amongst
these: $N_1=L_{\infty,0}$, $N_2=L_{0,\infty}$, $N_3=L_{1,1}$, $N_4=V(X-\omega
Z,Y-\omega^2 W)$, $N_5=V(X-\omega^2 Z, Y-\omega W)$. Interchanging $W$
with $X$ and $Y$ with $X$ we see that the pairs of lines $D_t$
defined as follows give precisely all lines in $S_0$ that meet
$L_{0,0}$:
\begin{eqnarray*}
	D_0 & = & L_{0,\infty}\cup L_{0,1} \\
	D_{\infty} & = & L_{\infty,0} \cup L_{1,0} \\
	D_1 & = & L_{1,1}\cup V(Y-W,Z+X-Y) \\
	D_{\omega} & = &
	V(W-\omega Y,Z-\omega^2 X)\cup
	V(W-\omega Y,Z+\omega^2 X-Y)\\
	D_{\omega^2} & = &
	V(W-\omega^2 Y,Z-\omega X)\cup
	V(W-\omega^2 Y,Z+\omega X-Y)
\end{eqnarray*}
We see that the 5 lines $N_i$ listed previously are the \emph{only} lines
that meet both $L_{0,0}$ \emph{and} $L_{\infty,\infty}$. Now consider
the union $Q_{ijk}$ of the locus of all lines that meet $N_i$, $N_j$, $N_k$ for
some triple $\{i,j,k\}$ contained in $\{1,2,3,4,5\}$. As seen
earlier this surface $Q_{ijk}$ is a smooth quadric surface which has
two rules, one given by the lines that meet $N_i$, $N_j$ and $N_k$
and another ruling that contains these lines. In particular, the
intersection of $Q_{ijk}$ and $S_0$ is a degree 6 locus and must
consist of 3 lines from the first ruling; $L_{0,0}$ and
$L_{\infty,\infty}$ give two such lines and we obtain one more which
we call $M_{ijk}$. Conversely, given a line $M$ inside $S$ which does
not meet $L_{0,0}$ or $L_{\infty,\infty}$, we can form a quadric $Q$
that is the union of all lines meeting $L_{0,0}$, $L_{\infty,\infty}$
and $M$. The intersection of $Q$ and $S$ is of degree $6$ and
contains these lines; thus it consists of exactly 3 lines out of
the $N_i$'s. We now see that we have $\binom{5}{3}=10$ lines $M$ that
are skew to each of $L_{0,0}$ and $L_{\infty,\infty}$. Combining the
above calculations we see that we have $9+5+(5-2)+10=27$ lines.

The following combinatorial description helps us to identify this as
a ``double-six''. Consider the 6-tuples
\begin{multline*}
	\calS_1=(P_0,P_1,P_2,P_3,P_4,P_5)\\
	=\left( L_{\infty,\infty}, L_{0,1}, L_{1,0},
	 V(Y-W,Z+X-Y),\right.\\
	 \left. V(Y-\omega W,Z+\omega^2 X-Y),
	 V(Y-\omega^2 W,Z+\omega X-Y) \right)
\end{multline*}
and
\begin{multline*}
	\calS_2=(Q_0,Q_1,Q_2,Q_3,Q_4,Q_5)\\
	=\left( L_{0,0}, L_{\infty,1}, L_{1,\infty},
	 V(Z-X,Y+W-Z),\right.\\
	 \left. V(Z-\omega X,Y+\omega^2 W-Z),
	 V(Z-\omega^2 X,Z+\omega W-Z) \right)
\end{multline*}
We observe that each of these 6-tuples consists of mutually disjoint
lines. We have already noted that $P_0$ (respectively $Q_0$) meets each
of the lines $Q_i$ (respectively $P_i$) for $i\neq 0$. Consider the
line $P_1$. The span of $P_1$ and $Q_0$ is a plane that contains
$N_1$. So $P_1$ meets $N_1$ which lies in the plane spanned
by $P_0$ and $Q_1$. On the other hand $P_1$ is disjoint from the line
$N_i$ for $i\neq 1$ and from the line $P_0$. The plane spanned by
these contains the line $Q_i$ and so $P_1$ must meet it. In other
words, $P_1$ meets all the elements of $\calS_2\setminus\{Q_1\}$.
Similarly, we can show that $P_i$ meets all the elements of
$\calS_2\setminus\{Q_i\}$ for other $i$.
\begin{quote}
	A \emph{double-six} consists of a pair of ordered 6-tuples of
	lines $(P_0,\dots,P_6)$ and $(Q_0,\dots,Q_6)$ such that each
	$P_i$ meets all the $Q_j$ except $Q_i$.
\end{quote}
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 $6+6+\binom{6}{2}=27$
lines.

The projective space of dimension $\binom{3+3}{3}-1=19$ parametrises
all homogeneous cubic polynomials in 4 variables or equivalently cubic
surfaces in $\bbP^3$. As we have seen before there is a 4 dimensional
variety $\bbG(1,\bbP^3)$ that parametrises lines in $\bbP^3$. Let $I$
denote the sub-variety of $\bbP^{19}\times\bbG(1,\bbP^3)$ 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 $\bbG(1,\bbP^3)$ is a linear $\bbP^{15}$ in
$\bbP^{19}$; so $I$ is also 19-dimensional. We have shown that the
fibre of $I\to\bbP^{19}$ consists of finitely many points over the
point corresponding to the surface $S_0$. It follows that this is a
surjective morphism. In particular, \emph{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. ({\bf 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, Koll\'ar, 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.

\end{document}