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Does any one know how to go from "Thom polynomials" to "Enumerating curves". I believe there is a relation between the two, but I don't know how to go about it. Let me make the question more concrete.

Suppose I want to find the number of degree d curves in $CP^2$, (passing through the right number of points so that you expect a finite answer), having a simple node ($A_1$ singularity). I know how to solve this problem, but I want to know a solution that uses the Thom polynomial.

In page 42 of this article

http://www.math.uic.edu/~jaca2009/notes/Weber.pdf

the Thom polynomials are calculated for quite a few singularities.

For $A_1$, the Thom polynomial is just 1. How does that give me the answer to my problem?

By the way the correct answer is $3(d-1)^2$.

Similarly, how many degree d curves are there in CP^2 that have an $A_3$ singularity? How would I do the problem using the Thom polynomial for $A_3$, which the author of the article has calculated.

The correct answer for the second question is $50d^2-192d+168$.

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1 Answer 1

The $A_1$ that you mention doesn't relate to an actual problem. You need to phrase the problem in terms of families of functions, and then it's the singularities of that family that correspond to certain geometrical phenomena. It's not the singularity type of the curve's equation. Let me try to explain.

One of the experts in Maxim Kazarian, no it's not me! But he gave a talk in one of our seminars a while back. I would recommend, for example looking at this paper. But before that. let me try to give a very rough outline of the ideas he gave in the seminar.

Let's say you wanted to know how many inflexions a generic curve, in the complex projective plane, of degree d has.

Set up some duel maps. For each point of a curve $C,$ consider the set of all unoriented lines passing through that point. This gives $C \times (\mathbb{CP}^2)^{\star}$, which is a submanifold of $\mathbb{CP}^2 \times (\mathbb{CP}^2)^{\star}$; where $\star$ denotes the duel. Next, consider the projection $\pi : C \times (\mathbb{CP}^2)^{\star} \twoheadrightarrow (\mathbb{CP}^2)^{\star}$ restricted to the submanifold $C \times (\mathbb{CP}^2)^{\star}$.

A point in $(\mathbb{CP}^2)^{\star}$ is a line, and its preimage under the projection $\pi : C \times (\mathbb{CP}^2)^{\star} \twoheadrightarrow (\mathbb{CP}^2)^{\star}$ are the points of $C$ that the line passes through. Then the fold points ($A_2$) correspond to ordinary tangencies and the pleat points ($A_3$) correspond to ordinary inflexions.

Next, look at the cohomology of the closure of the set of critical points of $\pi : C \times (\mathbb{CP}^2)^{\star} \twoheadrightarrow (\mathbb{CP}^2)^{\star}.$ This is where the singularity types come in. From the cohomology ring of this space we can determine polynomial relations. Depending on the degree, you only needed to work out the first few, say $d = 1,2$, to get the coefficients, and then know it for all $d$.

So to count the number of inflexions, you need to consider an $A_3$ singularity. To count the number of bitangent lines you need to consider the $A_2^2$ multi-singularity. But like I said, please look at Maxim Kazarian's webpage, and the paper that I linked to.

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