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Moduli Spaces of Polynomials in Two Variables
Javier Fernández de Bobadilla, Universiteit Utrecht, Netherlands
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Memoirs of the American Mathematical Society
2005; 136 pp; softcover
Volume: 173
ISBN-10: 0-8218-3593-9
ISBN-13: 978-0-8218-3593-7
List Price: US$66 Individual Members: US$39.60
Institutional Members: US\$52.80
Order Code: MEMO/173/817

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.

Graduate students and research mathematicians interested in algebraic geometry.

• Introduction
• Automorphisms of the affine plane
• A partition on $$\mathbb{C}[x,y]$$
• The geometry of the partition
• The action of Aut$$(\mathbb{C}^2)$$ on $$\mathbb{C}[x,y]$$
• The moduli problem
• The moduli spaces
• Appendix A. Canonical orders
• Bibliography