\documentclass{amsart}
\newcommand{\Oh}{{\mathcal O}}
\newcommand{\bbA}{{\mathbb A}}
\newcommand{\bbP}{{\mathbb P}}
\title{Introduction to the Algebraic Surfaces Workshop}
\author{Kapil Hari Paranjape}
\begin{document}
\maketitle
\section{The Beginning}
A number of different strands combined to become Algebraic Geometry as
we know it today.
\begin{itemize}
    \item The study of Riemann surfaces and algebraic curves. This was
        seen more algebraically as the study of Dedekind domains and a
        field of transcendence degree one. This study made contact with
        the study of algebraic number theory as initiated by Kummer and
        Kronecker.
    \item The study of Elliptic functions with Eisenstein series, the
        Weierstrass $\wp$-function, theta functions and so on led to the
        study of Abelian varieties and related varieties.
    \item Synthetic projective geometry went on the study of enumerative
        problems and the introduction of Grassmanians and other
        homogeneous varieties. It also got merged with the study of
        affine algebraic groups.
    \item The study of commutative algebra and homological algebra
        introduced a systematic method to prove all local (and some
        global) properties of algebraic varieties.
    \item The study of algebraic and differential topology. The relation
        between topological properties and differential invariants
        exemplified by (Say) the Gauss-Bonnet theorem. The study of Lie
        groups, especially matrix groups and their classifying spaces
        also merges in.
\end{itemize}
Most of you have seen some basics of the above concepts and have
probably studied algebraic curves in some detail. However, none of the
above topics by themselves exhibit the full glory of what algebraic
geometry has evolved into. (For example, one can get by with only a
little commutative algebra (Dedekind domains) when studying algebraic
curves.)

It is only when one starts studying Algebraic Surfaces that all of the
above strands come together in a way that none of the strands can be
extricated from the other.

\section{Some primary questions}
An algebraic surface can be defined as an irreducible, reduced algebraic
scheme $X$ of dimension 2 over a field $k$. Some people may instead ask
for a 2-dimensional complex manifold and some other (not fully
equivalent!) variants are possible as well.

With some basic knowledge of algebraic geometry one can immediately ask
many questions. For example,
\begin{itemize}
    \item Is every field of transcendence degree 2 associated with an
        algebraic surface? Is such a surface (if projective) unique? Can
        there be a non-quasi-projective surface associated with such a
        field?
    \item If a field of transcendence degree 2 is contained in the
        rational function field of two variables, is such a field
        automatically isomorphic to the rational function field of two
        variables? (Luroth's problem)
    \item Does the topology of an algebraic surface determine its
        ``type'' like in the case of curves? In other words, is there
        only one (connected) component for the moduli of algebraic
        structures for a fixed topological type? Are there
        surface ``types'' that cannot be found as hypersurfaces in
        projective three space?
    \item What are the possible homotopy types of algebraic
        surfaces? (Recall that the homotopy types of curves are rather
        limited. In particular, is there any restriction on what groups
        can be the fundamental groups of surfaces?
    \item The second homology of a surface will have a quadratic form
        (due to Poincare duality). What kinds of quadratic forms are
        possible. Are all classes in the second homology representable
        by algebraic curves on the surface?
    \item Is there any difference between surfaces in characteristic $p$
        and those in characteristic 0? Recall that the ``types'' of
        curves are the same in all characteristics.
    \item Is there a classification of surfaces similar to that for
        curves?
    \item Is there a theory of linear systems on surfaces analogous to
        the that for curves. Are there notions line Weierstrass points
        etc.? What is the minimal dimension of a projective space in
        which one can embed a projective algebraic surface.
\end{itemize}
During these two weeks we will attempt to look at some of these
questions and find suitable answers.
\section{Techniques}
We will use a number of ``standard'' algebraic tools during the lectures
that follow. These tools will be quickly recalled as we go along. In
addition, there are a number of techniques that we can utilise to study
surfaces.
\begin{description}
    \item[Geometric] We can realise a projective algebraic surface in a
        number of different ways: as a covering of the projective plane,
        as a family of curves parametrised by a curve, as a hypersurface
        in projective three space (or another three dimensional
        homogeneous space), as a variety dominated by a product of two
        curves.
    \item[Topological] We can study the topology of a surface by
        studying linear systems of curves on it, or by studying the
        properties of intersections of such curves. We can also study
        divisors in various linear (and non-linear) systems of curves on
        the surface. One important topological tool is that of a
        Lefschetz pencil which generalises the notion of a curve
        covering the projective line only simple ramification points.
    \item[Differential] In addition to 1-forms, we can also study
        holomorphic 2-forms on a surface. There are forms of various
        kinds giving rise to an algebraic version of the de Rham
        Theorem. The important complex algebraic technique, Hodge theory
        plays a much more significant role in the study of surfaces that
        in does in the theory of curves.
    \item[Birational] While smooth curves are determined by their
        function field, there are a number of smooth surfaces with the
        same function field. The study of how to go from one such
        surface from another (blowing-up and blowing-down) as well as
        the study of numerical invariants that are unchanged by such
        operations is a useful technique.
\end{description}
\section{Plan}
It is not possible to study surfaces in the linear order that one
studies curves or even abelian varieties. Thus, the approach that we
will follow is best explained by analogy with a music class! None of
the lecturers will attempt to give a ``complete'' proof of a theorem
(just as a music teacher does not give a concert during a class!).
Instead, we will tell you about the important things to watch for and
the main results to attain some grasp of (the ``vaadi'' and the
``samvaadi''!). We will have tutorials where (I hope!) some of you will
try some problems so that we can point out where you are taking a
convoluted path and where your short-cut will not work and why.

Note that strict linearity is not going to be maintained. So some
lectures will use concepts that will be explained in later lectures; you
will just have to suspend disbelief and move on. Each lecture will
introduce one or more concepts and explain them through examples and
through key results; they will occasionally highlight some key points in
the proofs of these result, but most often, especially in later
lectures, proofs will be skipped entirely or left for discussion in the
tutorial sessions. This is slightly different from the usual style of
lecturing in mathematics. Let's see how it works out!

\section{Mathematical introduction}
With that (rather too verbose for some!) introduction out of the way,
let us get to some mathematics.

An affine algebraic surface cab be defined as the ``locus of zeroes'' of a 
collection $f_1(x_1,\dots,x_p)$,\dots,$f_q(x_1,\dots,x_p)$ of polynomial
equations over a field $k$, so that the ring 
\[ R=\frac{k[x_1,\dots,x_p]}{<f_1,\dots,f_q>} \]
has the following properties:
 \begin{enumerate}
     \item $R$ is a domain.
     \item $R$ is two dimensional. This means that there is pair of
         elements $u_1$, $u_2$ in $R$ so that the sub-algebra
         $k[u_1,u_2]$ generated by them is a polynomial algebra (i.~e.\
         they are algebraically independent) and the inclusion
         $k[u_1,u_2]\to R$ is an integral (and separable) extension.
     \item Very often we will assume that $R$ is smooth over $k$, which
         means for each maximal ideal $m$ of $R$ we can choose $u_1$, $u_2$
         so that, the extension $k[u_1,u_2]\to R$ has the following
         properties:
            \begin{enumerate}
                \item The ideal in $R_m$ generated by $(u_1,u_2)$ is the
                    maximal ideal $mR_m$.
                \item The resulting field extension $k\to R/m$ is a
                    separable extension.
            \end{enumerate}
        When these two conditions are satisfied we say that $k[u_1,u_2]\to R_m$
        is \'etale.
 \end{enumerate}
Note that the field $K$ of quotients of $R$ is a finite separable
extension of $k(u_1,u_2)$. Conversely, given such an extension, we can
take $R$ to be the integral closure of $k[u_1,u_2]$ in $K$. One proves
(exercise!) that $R$ is a finitely generated $k$-algebra and (by
Hilbert's basis theorem) that the kernel ideal is generated by finitely
many polynomials.

The remaining question of interest is to ``compactify'' this surface.
For this, we pick a filtering $R_0=k\subset R_1\subset \dots$ of $R$ by
finite dimensional $k$ vector spaces $R_n$ so that $R_n\cdot R_m\subset
R_{m+n}$. For example, we can let $R_n$ by the image of polynomials of
degree at most $n$ in the variables $x_1$,\dots,$x_q$. We then form the
ring $S=\oplus_n R_n$ and note that it is a graded ring. If we have
chosen well(!), $S$ is generated by $R_1$, and so it is a quotient of
the polynomial ring as a graded ring. Hence,
    \[ S=\frac{k[X_0,X_1,\dots,X_p]}{<F_1,\dots,F_s>} \]
for some \emph{homogeneous} polynomials $F_1$, \dots, $F_s$. You can ask
yourself (exercise!) what the generators $X_i$ and whether you can
determine the polynomials $F_i$. The locus of zeroes of
$F_1$,\dots,$F_s$ is a projective algebraic surface containing our
original surface as an affine open sub-variety.

In general, one would like to think of an algebraic surface as ``made up
of affine surfaces by patching''. The question which remains is what
kind of patching we ``permit''. In the theory of schemes, we use the
Zariski topology for patching. If you are a complex geometry person you
may want to allow more general constructions. For example, we may allow
\'etale patching as was done by Moishezon. This sometimes gives us
compact complex surfaces that are not algebraic (and yet have a function
field of transcendence degree two). However, in the case of surfaces,
such surfaces \emph{have to be} singular. (In three or more dimensions
there are even smooth examples.) This already is a bit of a contrast
with the theory of curves where there are no such examples.

The ``simplest'' algebraic surfaces are  surely $\bbA^2$, the affine
plane and  $\bbP^2$, the projective plane. The fundamental formula for
(projective) plane curves is Bezout's theorem which says that a curve of
degree $m$ meets a curve of degree $n$ in $mn$ points \emph{if counted
properly}. In general, calculating the intersection of \emph{distinct}
curves in a surface follows the same approach. Things become interesting
when we want to calculate the (virtual) intersection number of a curve
with itself. In order to preserve linearity of such intersections, one
arrives at a canonical intersection number (at least for smooth
surfaces)---in some cases the number can even be negative; which may not
be surprising from a topological perspective.

It turns out that each curve $C$ in a smooth surface $X$ (or in the
smooth locus of a singular surface) gives rise to a line bundle $\Oh_X(C)$ 
on $X$ and hence a class $[C]=c_1(\Oh_X(C))$ (first Chern class of the
line bundle) in the second cohomology $H^2(X)$ of the surface. We can
therefore calculate the cap of this class with any other class in
$H^2(X)$. For the class $[D]$ of another curve in $X$, it turns out that
$[C]\cap[D]=(C\cdot D)[p]$ where $C\cdot D$ is the intersection number
calculated algebro-geometrically and $[p]$ is the class in $H^4(X)$ of a
(any) smooth point $p$ on $X$.

The above interplay between curves on a surface, line bundles on the
surface and the associated homology classes is a very interesting and
important aspect of the study of surfaces.

We have already mentioned the module of differentials $\Omega^1_{R/k}$
for a ring $R$. This patches up to give a coherent sheaf
$\Omega^1_{X/k}$ on a scheme $X$ over $k$. By taking exterior powers we
can form the differential graded algebra $\Omega^{\cdot}_{X/k}$ just as
we do for manifolds. When $X$ is smooth over $k$, this is exact for the
\emph{completed} localisation, but \emph{not} in general for in affine
open sets. Grothendieck, generalised the classical theory of
differential forms of the first and third kind and  pointed out that
this DGA can none-the-less be used to calculate the topological structure
of $X$. This result and Serre duality underline the importance of
studying 1-forms and 2-forms on a surface.

The projective plane is by no means the only surface whose field of
rational functions is isomorphic to $k(x,y)$. Take a rational
normal curve $C_a$ in $\bbP^a$ and a rational normal curve $C_b$ in
$\bbP^b$ and put $\bbP^a$ and $\bbP^b$ in a disjoint fashion as linear
subspaces in $\bbP^{a+b+1}$; moreover, pick an isomorphism between $C_a$
and $C_b$. The union of the lines that join pairs of corresponding
points on $C_a$ and $C_b$ gives a surface $F_{|a-b|}$, called a
Hirzebruch surface. By construction it is ``ruled''; in that it is a
union of lines. More generally, given any map of a curve $C$ to the
Grassmannian $G(1,N)$ of lines in $\bbP^N$ gives rise to a ruled surface
in $\bbP^N$. This is an important class of surfaces that we will study.
How about families of rational curves of higher degree? This is question
behind Tsen's theorem.

One natural generalisation of the theory of Elliptic curves is the study
of compact complex tori of dimension two. When such a surface has
non-trivial rational functions (which is not always!), then one can show
that it is a projective variety and that the group structure is
algebraically defined. This leads to the study of Abelian Surfaces
which has many aspects that are more detailed and intriguing than the
study of general Abelian varieties of higher dimensions.

In any detailed study of projective curves or of Riemann surfaces, we
are introduced to the Abelian variety called the Jacobian $J(C)$ of the
curve $C$. This is an \emph{algebraic} form of the group of line
bundles of degree 0 on the curve; it is also the ``initial object'' in
the category of abelian varieties admitting a map from $C$. In the case
of surfaces, these two varieties can be distinct and are called the
Picard and Albanese varieties of the surface. We will show some key ideas
behind the construction of these important invariants of a surface.

A different kind of generalisation of the notion of an elliptic curve is
the notion of a K3 surface. This is a simply connected surface which has
a global nowhere vanishing 2-form. It was proved by Kodaira that all
such surfaces have the same topological type; however the moduli space
is not as simple as one might think as there are K3 surfaces which are
not algebraic! The existence of K3 surfaces is what clearly indicates
that the classification of surfaces is much more complicated than that
of curves --- we have two generalisations of elliptic curves!

Another aspect of the study of surfaces is that singularities of
surfaces have a lot more topological information that just the bunching
together of points which happens on a curve. This leads to the
fascinating study of surface singularities; we begin with rational double
points, which is a kind of singularity that does not exist in dimension
one!

There are a few other topics that will be touched upon during the second
week that it is difficult to introduce at this point. All in all, we are
doing our best to throw as much of surface theory that we (the speakers)
have some handle on. We hope you will catch some of these throws and
get infected with the enjoyment of this fascinating subject.

\end{document}