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:
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 twisted cubic curve is the zero locus of three polynomials:
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 - bundles over . Sometimes it is important to classify all such bundles over a fixed variety - this describes additional invariants of , which may be used for instance to identify two non-isomorphic varieties.
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 subvariety of . (The word ``nice'' has a very technical meaning, usually called ``flatness'' of families. We shall not discuss this here since it will take us too far afield.)
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 parameter spaces.
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, non-reduced) hypersurfaces is in fact a subvariety of the corresponding parameter space .
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.