The geometry of stable quotients in genus one

Abstract

Stable quotients provide an alternative to stable maps for compactifying spaces of maps. When \(n \ge 2\), the space \(\overline{Q}_{g}({\mathbb {P}}^{n-1},d) = \overline{Q}_{g}(G(1,n),d)\) compactifies the space of degree \(d\) maps of smooth genus \(g\) curves to \({\mathbb {P}}^{n-1}\), while \(\overline{Q}_{g}(G(1,1),d) \simeq \overline{M}_{1, d \cdot \epsilon }/S_d\) 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 \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\), for all \(n \ge 1\). We show these schemes are projective, unirational, and have Picard number 2. Then we give generators for the Picard group, compute the canonical divisor, the cones of ample divisors, and in the case \(n=1\) the cones of effective divisors. We conclude that \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\) is Fano if and only if \(n(d-1)(d+2) < 20\). Moreover, we show that \({\overline{Q}}_{1}({\mathbb {P}}^{n-1},d)\) 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 Poincaré polynomial.

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Acknowledgments

The author would like to thank the following people. My advisor R. Pandharipande for patiently teaching me the techniques used in this paper. I. Coskun for the course “The birational geometry of the moduli spaces of curves” he gave at the School on Birational Geometry and Moduli Spaces June 2010 which inspired me to consider the questions about the cones of nef and effective divisors. The author would also like to thank O. Biesel, D. Chen, A. Deopurkar, M. Fedorchuck, C. Fontanari, J. Kollar, J. Li, D. Oprea, A. Patel, S. Patrikis, A. Pixton, D. Ross, V. Shende, D. Smyth, R. Vakil, M. Viscardi, M. Woolf, and A. Zinger for helpful conversations and the anonymous referee for many helpful suggestions and corrections. The author was supported by an NSF graduate fellowship.

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Correspondence to Yaim Cooper.

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Cooper, Y. The geometry of stable quotients in genus one. Math. Ann. 361, 943–979 (2015). https://doi.org/10.1007/s00208-014-1079-6

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