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.
Readership
Graduate students and research mathematicians interested in
algebraic geometry.