__ Example 1.__ Let be the (smooth)
quadric surface in
given as the zero locus
of one homogeneous quadratic equation:

The quadric consists of two families of lines, each of which sweeps on its own:

and

In terms of the matrix

More generally, for any two varieties and , there is a (unique) variety with projection maps and , the fibers of which are correspondingly copies of and of . The uniqueness of such a variety is ensured by some extra (natural) properties coming from category theory - such properties are exactly what we would expect from a variety deserving to be called the ``product'' of two varieties.

The actual construction of this product is given by a
(seemingly random) map, called the *Segre embedding*. For starters, to
construct the product
, we define the
Segre map by

Now it is not hard to construct the product of any two
varieties and : if
and
, then is the subvariety of
, satisfying the extra equations
coming from and .

__ Example 2.__ The

The twisted cubic lies on the quadric surface , and is a curve of type on , i.e. meets every line in one family in 2 points, and every line in the other family - in 1 point (prove this!) Prove also that the zero locus of any two of the three quadratic polynomials defining is the union of and a line on meeting in two points (or being tangent to ). How many such tangent lines to are there in the family consisting generically of lines meeting in two points?

An alternative description of is the image of the *Veronese embedding* of
in
:

__ Example 3.__ What are the hyperplane sections
of , i.e.
, as the hyperplane
varies in
?

The strategy here is to restrict the equation of to the hyperplane , and to realize that a homogeneous quadratic polynomial in 3 variables is either irreducible (smooth plane conic hyperplane section of ), or factors as the product of two homogeneous linear factors (the hyperplane section here is the union of two intersecting lines in ). Prove that we will never get the quadratic polynomial to factor as a perfect square of a linear form (i.e. no hyperplane in intesects in a ``double line'').

We push the above considerations an inch further to ask the
following question: can we construct a variety
which
in some reasonable way will be the family of *all* hyperplane
sections of ? In other words, can we separate all hyperplane
sections of , so that they do not intersect anymore, but stay as
fibers of some map? The answer is ``YES'', yet we have to work a bit
to construct this variety.

For starters, we can construct the *universal hyperplane*
in
- apriori, we want this to be a variety,
representing *all* hyperplanes in
, in other words,
should be a family of all hyperplanes in
.
The first step is to realize what variety *parametrizes* these
hyperplanes - this is the *dual*
,
which is really
all over again, by with different
coordinates. If
has coordinates
,
then a hyperplane in
is given by a linear form
for some fixed
. Thus, the dual
has coordinates
; points in
correspond to
hyperplanes in
, and hyperlanes in
correspond to points in
.

Now, the universal hyperplane
should be defined
as a set by

Finally, to construct the *universal hyperplane section*
of the quadric , we only have to intersect with
:

Maps of varieties whose fibers are all isomorphic to some are called

__ Example 4.__ As we saw above, all
hyperplanes in
can be parametrized by the variety
(which is isomorphic to
.) In this
case, the points of
are in 1-1 correspondence with
the hyperplanes in question, and
reflects (in a
certain sense) how the hyperplanes vary in
- that is,
for any ``nice'' family
of hyperplanes in
the subset of
corresponding to
is a

One can easily generalize the above construction to parametrize all hyperplanes in by the dual projective space . A natural question arises: Can we find varieties parametrizing other objects, say, conics in ? Such varieties are called

Let be the set of all conics in . If we fix the coordinates of to be , then a conic is determined upto a scalar by a quadratic equation:

Similarly, the parameter space of all hypersurfaces of degree in (i.e. subvarieties given by single degree homogeneous polynomials on ) is where . Note a slight technicality here: we have included as points in ``hypersurfaces'' corresponding to polynomials with multiple factors. For example, in the case of conics in , we included as points in all ``double'' lines. One can show that the set of such ``multiple'' (or more precisely,

__ Example 5.__ Parameter spaces
parametrize usually not just objects sharing some common
properties, but also the embeddings of in projective space.
For example, there are really only three types of conics in
- the irreducible (smooth) conics, the joins of
two different lines, and the double lines. Every irreducible conic
can be transformed into any other irreducible conic after a suitable
change of variables (coordinate change) on
, etc.
Thus, in constructing
, we grossly ``overcounted''
the irreducible conics (well, we were parametrizing, therefore,
not just the conics, but the pairs where is a plane
conic and
is an embedding.)

The philosophy of viewing a variety as an object with a given
embedding in some
is inherent to XIX century algebraic
geometry, especially to the Italian school. XX century changed this
view by considering varieties as objects on their own, disregarding
particular embeddings in projective space. For example, any
irreducible conic in
is really a
embedded in a certain way in
:

Some *extrinsic* properties, however, change, and these
cause the different embeddings of
to look different.
For example, define the *degree* of
to be the number of points in the intersection of a
general hyperplane in
with . Thus, the conics in
have degree , and will keep their degree if we embed
now
as a linear subspace of a bigger
.
However, the twisted cubic in
has degree (one
way to see this is to recall that a line in one ruling meets in 1
point, while a line in the other ruling meets in 2 points.)

While parameter spaces may take into account such extrinsic
properties as *degrees* of varieties, *moduli spaces* usually
parametrize objects according to only their intrinsic properties, and
hence are much harder to be constructed. To even state what common
instrinsic properties can be characterized will take too much ink on
this handout. But let us mention one very famous example - the
moduli space
of *smooth curves of genus *.
^{1} These curves do not lie (and cannot be embedded in
general) in the same projective space
. The best we can
say is that each such (non-hyperelliptic) curve can be embedded in
, but we don't care about these embeddings anyways.
Yet,
can be constructed, and it is a variety of
dimension for . For there is only one such
curve -
, so
is really just one point;
for - the elliptic curves can be effectively parametrized by a
certain cross ratio, and hence
.

A further development of this theory is the *Deligne-Mumford* compactification of
. Since
is *not* a projective variety, one can have a
nice family of *smooth curves* degenerating to a singular curve,
but
does not have any points to reflect the limiting
singular member of the family. The question arises - what is the
``minimal'' set of *singular curves* must be added to the set of
smooth curves in order to obtain a ``nice'' moduli space
, compactifying
? Deligne
and Mumford chose (and for very good reasons) the set of the
so-called *stable* curves of genus . These are connected
curves with at most *nodal* type of singularities (e.g. take two
lines intersecting in
), and such that if they contain a
-component, then the latter must meet at least 3 other
components of the curve. The last condition is added to ensure that
the curves have finite groups of automorphisms. With this said,
is *the moduli space of all stable
curves of genus *. It is a projective variety which contains
as an open dense set, and it reflects naturally the
variation of ``nice'' families of stable curves. Moreover, any
``nice'' family whose general members are smooth curves, but whose
special members can be as nasty as you wish, can be brought in an
essentially one way to a family with only stable members. This
process is called *semistable reduction* and it is the basis for
many related constructions in algebraic geometry.