Compactifying the space of stable maps

Compactifying the state space for alternative theories of gravity

In this paper we address important issues surrounding the choice of variables when performing a dynamical systems analysis of alternative theories of gravity. We discuss the advantages and disadvantages of compactifying the state space, and illustrate this using two examples. We first show how to define a compact state space for the class of LRS Bianchi type I models in R-gravity and compare to...

Locally Stable Maps of the 3-Sphere into 4-Space

We classify locally stable maps from the 3-sphere to 4-space up to homotopy through locally stable maps. The equivalence class of a locally stable map f is determined by the isotopy type τ(f) of its framed singularity link, its generalized normal degree ν(f), and the algebraic number of cusps κ(f) of any generic extension of f to a map of the 4-disk into 5-space. Furthermore, relations between ...

Towards Mori’s Program for the Moduli Space of Stable Maps

We introduce and compute the class of a number of effective divisors on the moduli space of stable maps M0,0(P,d), which, for small d, provide a good understanding of the extremal rays and the stable base locus decomposition for the effective cone. We also discuss various birational models that arise in Mori’s program, including the Hilbert scheme, the Chow variety, the space of k-stable maps, ...

Relative Compactness and Product Stable Quotient Maps

A Compactification of the Moduli Space of Self-maps of Cp via Stable Maps

We present a new compactification M(d, n) of the moduli space of self-maps of CP of degree d with n markings. It is constructed via GIT from the stable maps moduli space M0,n(CP 1 ×CP, (1, d)). We show that it is the coarse moduli space of a smooth Deligne-Mumford stack and we compute its rational Picard group.