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The Effective Cone of the Kontsevich Moduli Space

Published online by Cambridge University Press:  20 November 2018

Izzet Coskun ,
Joe Harris  and
Jason Starr
Izzet Coskun
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607. e-mail: coskun@math.uic.edu
Joe Harris
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794. e-mail: jstarr@math.sunysb.edu
Jason Starr
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138. e-mail: harris@math.harvard.edu
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Abstract

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In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, ${{\overline{M}}_{0,0}}\left( {{\mathbb{P}}^{r}},\,d \right)$ , stabilize when $r\,\ge \,d$ . We give a complete characterization of the effective divisors on ${{\overline{M}}_{0,0}}\left( {{\mathbb{P}}^{d}},\,d \right)$ . They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

Keywords

14D2014E9914H10

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 4 , 01 December 2008 , pp. 519 - 534
Copyright
Copyright © Canadian Mathematical Society 2008

References

[ACGH] Arbarello, E., Cornalba, M., Griffiths, P. A., and Harris, J., Geometry of Algebraic Curves. Vol. I. Grundlehren der Mathematischen Wissenschaften 267, Springer-Verlag, New York, 1985.Google Scholar
[CM] Ciliberto, C. and Miranda, R., Degenerations of planar linear systems. J. Reine Angew. Math. 501(1998), 191–220.Google Scholar
[C] Coskun, I.. Degenerations of surface scrolls and the Gromov-Witten invariants of Grassmannians. J. Algebraic Geom. 15(2006), no. 2, 223–284.Google Scholar
[CS] Coskun, I. and Starr, J., Effective divisors on the space of maps to Grassmannians. Int. Math. Res. Not. 2006, Art. ID 35273.Google Scholar
[EH] Eisenbud, D. and Harris, J., The Kodaira dimension of the moduli space of curves of genus ≥ 23 . Invent. Math. 90(1987), no. 2, 359–387.Google Scholar
[Far1] Farkas, G., Syzygies of curves and the effective cone of Mg . Duke Math. J. 135(2006), no. 1, 53–98.Google Scholar
[Far2] Farkas, G.. is of general type. http://www-irm.mathematik.hu-berlin.de/_farkas/m22.pdf. Google Scholar
[Far3] Farkas, G.. Koszul divisors on moduli spaces of curves. To appear, Am. J. Math.Google Scholar
[FaP] Farkas, G. and Popa, M., Effective divisors on , curves on K3 surfaces, and the slope conjecture. J. Algebraic Geom. 14(2005), no. 2, 241–267.Google Scholar
[FP] Fulton, W. and Pandharipande, R., Notes on stable maps and quantum cohomology. In: Algebraic Geometry. Proc. Sympos. Pure Math. 62, American Mathematical Society, Providence, RI, 1997, pp. 45–96.Google Scholar
[H] Harris, J., On the Kodaira dimension of the moduli space of curves. II. The even-genus case. Invent. Math. 75(1984), no. 3, 437–466.Google Scholar
[HMo1] Harris, J. and Morrison, I., Slopes of effective divisors on the moduli space of stable curves. Invent. Math. 99(1990), no. 2, 321–355.Google Scholar
[HMo2] Harris, J. and Morrison, I., Moduli of Curves. Graduate Texts in Mathematics 187, Springer-Verlag, Berlin, 1998.Google Scholar
[HM] Harris, J. and Mumford, D., On the Kodaira dimension of the moduli space of curves. With an appendix by William Fulton. Invent. Math. 67(1982), no. 1, 23–88.Google Scholar
[Ha] Hartshorne, R., Algebraic Geometry. Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977.Google Scholar
[Kh] Khosla, D., Moduli space of curves with linear series and the slope conjecture. arXiv:math/0608024v1.Google Scholar
[Pa] Pandharipande, R.. Intersections of Q -divisors on Kontsevich's moduli space and enumerative geometry. Trans. Amer. Math. Soc. 351(1999), no. 4, 1481–1505.Google Scholar
[V] Vakil, R., The enumerative geometry of rational and elliptic curves in projective space. J. Reine Angew. Math. 529(2000), 101–153.Google Scholar
[Ya] Yang, S., Linear systems in P 2 with base points of bounded multiplicity. arXiv:math.AG/0406591.Google Scholar