I have read the book 'Enumerative Geometry and String Theory' by Katz, and it left me with some questions. It is outlined in the text how ideas from String theory and TQFT has enriched enumerative geometry. Examples of problems that couldn't be solved, but were solved using these methods are:
- 'Number' of rational curves of degree $d$ in the quintic threefold
- Number of degree $d$ curves passing through $3d-1$ points in $\mathbb{P}^2$.
Personally, the fact that the second problem with $d\geq 5$ couldn't be solved without the help of physics was mind-boggling, especially when the answer is as small as 87,304. The classical methods outlined in the text seem pretty strong, and I want to know what exactly was the difficulty was to solving these problems.
One point that Katz mentions is that the moduli space of stable maps was motivated by physics. But historically Deligne and Mumford compactified similar moduli spaces, and why couldn't people use this moduli space to solve the problems?
Please note that I am not fully aware of some 'subtleties' that might be present in enumerative geometry, since I know only very crudely how enumerative problems are solved. If you could identify the hard point inside(or outside!) the follwing steps of solving enumerative problems that would be nice.
- Find a moduli space compactifying the geometric objects considered.
- Compute the cohomology ring of the moduli space.
- Identify what needs to be integrated over the moduli spade.
- Find the excess intersection, and show that the answer is actually enumerative.