MODULI SPACES OF POLYNOMIALS IN TWO VARIABLES by Javier Fernandez de Bobadilla ( 9780821835937 )

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Title:

MODULI SPACES OF POLYNOMIALS IN TWO VARIABLES

By:

Format:

Paperback

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ISBN 10:

0821835939

ISBN 13:

9780821835937

Publisher:

AMERICAN MATHEMATICAL SOCIETY

Pub. date:

15 January, 2005

Series:

Memoirs of the American Mathematical Society No. 173

Pages:

136

Description:

Investigates the geometry of the orbit space. This book associates a graph with each polynomial in two variables that encodes part of its geometric properties at infinity. It also defines a partition of $\mathbb{C}[x,y]$ imposing that the polynomials in the same stratum are the polynomials with a fixed associated graph.

Synopsis:

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.

Publication:

Imprint:

American Mathematical Society

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