Mathematics – Algebraic Geometry
Scientific paper
2011-03-04
Mathematics
Algebraic Geometry
Typos corrected. Reference [3] added
Scientific paper
A quasi-map to P^{n-1} refers to a line bundle on a quasi-stable pointed curve together with n ordered sections. It was proved by H.-L. Chang and J. Li that quasi-maps of degree d>0 to P^{n-1} over m-pointed curves of genus g form an algebraic stack and that any open Deligne-Mumford substack has a perfect obstruction theory. Therefore a proper separated Deligne-Mumford open substack admits a virtual fundamental class on which curve counting invariants are defined as intersection numbers. Examples include the moduli stack of stable maps and the moduli stack of stable quotients. In this paper, we introduce the notion of delta-stable quasi-maps and show that the open substack of delta-stable quasi-maps is a proper separated Deligne-Mumford stack for each value of the stability parameter delta>0 except for a finite set of walls. We also consider the GSW model for delta-stable quasi-maps to P^4 with p-fields and obtain invariants. When delta is close to 0 and m=0, the moduli of delta-stable quasi-maps admits a forgetful morphism to Caporaso's moduli space of balanced line bundles. The wall crossings are shown to be contraction morphisms from larger delta to smaller.
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