The Geometry of Stable Quotients in Genus One

Abstract

Stable quotients provide an alternative to stable maps for compactifying spaces of maps. When n > 1, the space Qbar_g(P^{n-1}, d) = Qbar_g(G(1, n), d) compactifies the space of degree d maps of smooth genus g curves to P^{n-1}, while Qbar_g(G(1, 1), d) = M_{1,d.e}/S_d is a quotient of a Hassett weighted pointed space. In this paper we study the coarse moduli schemes associated to the smooth proper Deligne-Mumford stacks Qbar_1(P^{n-1}, d), for all n > 0. We show these schemes are projective, unirational, and have Picard number 2. Then we give generators for the Picard group, compute the canonical divisor, and the cones of ample and effective divisors. We conclude that Qbar_1(P^{n-1},d) is Fano if and only if n(d- 1)(d + 2) < 20. Moreover, we show that Qbar_1(P^{n-1},d) is a Mori Fiber space for all n; d, hence always minimal in the sense of the minimal model program. In the case n = 1, we write in addition a closed formula for the Poincare polynomial.