Abstract
We generalize Keel’s results on the intersection theory of \({\overline{M}}_{0,n}\) to the first nontrivial higher-dimensional case, the stable pair compactification \({\overline{M}}(3,6)\) of the moduli space M(3, 6) of six lines in the plane. In particular, we describe a sequence of blowups yielding a small resolution of \({\overline{M}}(3,6)\), and use this sequence to give a presentation of the Chow ring of any small resolution of \({\overline{M}}(3,6)\), analogous to Keel’s presentation of \(A^*({\overline{M}}_{0,n})\). We also introduce tautological classes on the moduli space of stable hyperplane arrangements and discuss their intersection theory on \({\overline{M}}(3,6)\). As an application of our results, we give an independent proof of Luxton’s result that \({\overline{M}}(3,6)\) is the log canonical compactification of M(3, 6).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alexeev, V.: Weighted Grassmannians and stable hyperplane arrangements (2008). arXiv:0806.0881
Alexeev, V.: Moduli of Weighted Hyperplane Arrangements. Advanced Courses in Mathematics, CRM Barcelona, Springer, (2015)
Aluffi, P.: Chern classes of blow-ups. Math. Proc. Cambridge Philos. Soc. 148(2), 227–242 (2010)
Corey, D.: Initial degenerations of Grassmannians. Sel. Math. New Ser. 27(4), 57 (2021). https://doi.org/10.1007/s00029-021-00679-6
Faber, C.: Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians. In: Hulek, K., et al. (eds.) New Trends in Algebraic Geometry. London Mathematical Society Lecture Note Series, vol. 264, pp. 93–109. Cambridge University Press (1999)
Fulton, W.: Intersection Theory. 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 2. Springer, Berlin (1998)
Fulton, w., MacPherson, R, : A compactification of configuration spaces. Ann. Math. 139(1), 183–225 (1994)
Gallardo, P., Routis, E.: Wonderful compactifications of the moduli space of points in affine and projective space. Eur. J. Math. 3(3), 520–564 (2017)
Getzler, E.: Topological recursion relations in genus \(2\). In: Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), pp. 73–106. World Scientific, River Edge (1998)
Hacking, P., Keel, S., Tevelev, J.: Compactification of the moduli space of hyperplane arrangements. J. Algebraic Geom. 15(4), 657–680 (2006)
Hacon, C.D., Kovács, S.J.: Generic vanishing fails for singular varieties and in characteristic \(p>0\). In: Hacon, C.D., et al. (eds.) Recent Advances in Algebraic Geometry. London Mathematical Society Lecture Note Series, vol. 417, pp. 240–253. Cambridge University Press, Cambridge (2015)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)
Hassett, B.: Moduli spaces of weighted pointed stable curves. Adv. Math. 173(2), 316–352 (2003)
Herrmann, S., Jensen, A., Joswig, M., Sturmfels, B.: How to draw tropical planes. Electron. J. Comb. 16 (2009). http://www.combinatorics.org/Volume_16/Abstracts/v16i2r6.html
Herrmann, S., Joswig, M., Speyer, D.: Dressians, tropical Grassmannians, and their rays. Forum Math. 26(6) (2014)
Kapranov, M.M.: Chow quotients of Grassmannians. I. In: Gelfand, S., Gindikin, S. (eds.) M. Gel’fand Seminar Advances in Soviet Mathematics, vol. 16.2, pp. 29–110. American Mathematical Society, Providence (1993)
Keel, S.: Intersection theory of moduli space of stable \(n\)-pointed curves of genus zero. Trans. Amer. Math. Soc. 330(2), 545–574 (1992)
Keel, S., McKernan, J.: Contractible extremal rays on \({\overline{M}}_{0,n}\). In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli, vol. II, pp. 115–130. International Press, Somerville (2013)
Keel, S., Tevelev, J.: Geometry of chow quotients of grassmannians. Duke Math. J. 134(2), 259–311 (2006)
Kimura, S.: Fractional intersection and bivariant theory. Comm. Algebra 20(1), 285–302 (1992)
Knudsen, F.F.: The projectivity of the moduli space of stable curves, II: The stacks \({{M}}_{g, n}\). Math. Scand 52(2), 161–199 (1983)
Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)
Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix airy function. Comm. Math. Phys. 147(1), 1–23 (1992)
Li, L.: Wonderful compactification of an arrangement of subvarieties. Michigan Math. J. 58(2), 535–563 (2009)
Luxton, M.A.: The Log Canonical Compactification of the Moduli Space of Six Lines in \({\mathbb{P}}^{2}\). Ph.D Thesis. The University of Texas at Austin (2008)
Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Shafarevich, I.R. (ed.) Arithmetic and Geometry, Vol. II. Progress in Mathematics, vol. 36, pp. 271–328. Birkhäuser, Boston (1983)
Petersen, D.: The Chow ring of a Fulton-MacPherson compactification. Michigan Math. J. 66(1), 195–202 (2017)
Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)
Witten, E.: Two-dimensional gravity and intersection theory on moduli space. In: Surveys in Differential Geometry (Cambridge, MA, 1990), pp. 243–310. Lehigh University, Bethlehem (1991)
Acknowledgements
The author is grateful for the help and support of his advisor, Valery Alexeev.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was partially supported by the NSF Grant DMS-1902157.
Rights and permissions
About this article
Cite this article
Schock, N. Intersection theory of the stable pair compactification of the moduli space of six lines in the plane. European Journal of Mathematics 8, 139–192 (2022). https://doi.org/10.1007/s40879-021-00508-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-021-00508-2