Moduli Spaces of Polynomials in Two Variables (eBook)

Javier Fernandez de Bobadilla (Autor)

eBook Download: PDF

136 Seiten
American Mathematical Society (Verlag)
978-1-4704-0418-5 (ISBN)

Infos

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.

Produkt-Details

• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• ISBN-10: 1-4704-0418-4 / 1470404184
• ISBN-13: 978-1-4704-0418-5 / 9781470404185