Moduli Spaces of Polynomials in Two Variables
2005
American Mathematical Society (Verlag)
978-0-8218-3593-7 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-3593-7 (ISBN)
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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.
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.
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.
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.
| Erscheint lt. Verlag | 1.8.2005 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Zusatzinfo | illustrations |
| Verlagsort | Providence |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| ISBN-10 | 0-8218-3593-9 / 0821835939 |
| ISBN-13 | 978-0-8218-3593-7 / 9780821835937 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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