# Moduli Spaces of Polynomials in Two Variables

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*Javier Fernández de Bobadilla*

In the space of polynomials in two variables \(\mathbb{C}[x,y]\) with
complex coefficients we let the group of automorphisms of the affine plane
\(\mathbb{A}^2_{\mathbb{C}}\) act by composition on the right. In this
paper we investigate the geometry of the orbit space.

We associate a graph with each polynomial in two variables that encodes
part of its geometric properties at infinity; we define a partition of
\(\mathbb{C}[x,y]\) imposing that the polynomials in the same stratum
are the polynomials with a fixed associated graph. The graphs associated with
polynomials belong to certain class of graphs (called behaviour
graphs), that has a purely combinatorial definition. We show that any
behaviour graph is actually a graph associated with a polynomial. Using this we
manage to give a quite precise geometric description of the subsets of the
partition.

We associate a moduli functor with each behaviour graph of the class, which
assigns to each scheme \(T\) the set of families of polynomials with the
given graph parametrized over \(T\). Later, using the language of
groupoids, we prove that there exists a geometric quotient of the subsets of
the partition associated with the given graph by the equivalence relation
induced by the action of Aut\((\mathbb{C}^2)\). This geometric
quotient is a coarse moduli space for the moduli functor associated with the
graph. We also give a geometric description of it based on the combinatorics
of the associated graph.

The results presented in this memoir need the development of a certain
combinatorial formalism. Using it we are also able to reprove certain known
theorems in the subject.

#### Table of Contents

# Table of Contents

## Moduli Spaces of Polynomials in Two Variables

- Contents v6 free
- Introduction vii8 free
- Chapter 1. Automorphisms of the affine plane 112 free
- Chapter 2. A partition on C[x,y] 1324
- Chapter 3. The geometry of the partition 2940
- Chapter 4. The action of Aut(C[sup(2)]) on C[x,y] 6172
- Chapter 5. The moduli problem 7788
- Chapter 6. The moduli spaces 89100
- Appendix A. Canonical orders 127138
- Bibliography 135146