The basic question of enumerative geometry can be simply stated
as:
How many geometric objects of a given type satisfy given geometric
conditions?
For instance, one may ask for
- the number of lines intersecting 4 fixed lines in P^3 (the answer is 2);
- the number of lines on a smooth
cubic surface in P^3 (the
answer is 27);
- the number of lines on a quintic threefold in P^4 (the
answer is 2,875);
- the number of conics on a quintic threefold (the answer is
609,250);
- ...
In many cases, this comes down to calculating intersection
numbers
on moduli spaces.
In the first part of the course, we will discuss Chern classes and show
some of their applications to enumerative questions.
In the second part, I will give an introduction to the moduli space of
curves. In particular, I hope to include a discussion of the Quot scheme,
of geometric invariant theory, outline the construction of the moduli
space, and possibly calculate one or two concrete intersection
numbers.
(This may be a bit too ambitious for a one quarter course, so the
goals
may change slightly as we go.)