References Cited. [4] A. Geraschenko, S. Morrison, and R. Vakil, Mathoverflow, Notices of the Amer. Math. Soc. 57 no. 6, June/July 2010, 701.

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1 References Cited [1] S. Billey and R. Vakil, Intersections of Schubert varieties and other permutation array schemes, in Algorithms in Algebraic Geometry, IMA Volume 146, p , [2] I. Coskun and R. Vakil, Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus, in Algebraic Geometry Seattle 2005 Part 1, , Proc. Sympos. Pure Math., 80, Amer. Math. Soc., Providence, RI, [3] I. P. Goulden, D. M. Jackson, and R. Vakil, A short proof of the λ g -conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves, J. Reine Angew. Math. (Crelle s Journal) 637 (2009), [4] A. Geraschenko, S. Morrison, and R. Vakil, Mathoverflow, Notices of the Amer. Math. Soc. 57 no. 6, June/July 2010, 701. [5] I.P. Goulden, D.M. Jackson, and R. Vakil, The moduli space of curves, double Hurwtiz numbers, and Faber s intersection number conjecture, Annals of Combinatorics, to appear. [6] B. Howard, J. Millson, A. Snowden, and R. Vakil, A new description of the outer automorphism of S 6, and the invariants of six points in projective space, J. Comb. Theory Ser. A, 115 (2008), no. 7, [7] B. Howard, J. Millson, A. Snowden, and R. Vakil, The relations among invariants of points on the projective line, C.R. Math. Acad. Sci. Paris 347 no , Oct. 2009, [8] B. Howard, J. Millson, A. Snowden, and R. Vakil, The equations for the moduli space of n points on the line, Duke Math. J. 146 no. 2 (2009), [9] B. Howard, J. Millson, A. Snowden and R. Vakil, The ideal of relations for the ring of invariants of n points on the line, J. Eur. Math. Soc., to appear (earlier version arxiv: ). [10] B. Howard, J. Millson, A. Snowden, and R. Vakil, The ideal of relations for the ring of invariants of n points on the line: integrality results, submitted. [11] B. Howard, J. Millson, A. Snowden and R. Vakil, The geometry of eight points in projective space: representation theory, Lie thery, dualities, in preparation (expected on arxiv fall 2010). [12] J. Keller and R. Vakil, π p, the value of π in l p, Amer. Math. Monthly, 116 (Dec. 2009), no. 10,

2 [13] Y.P. Lee and R. Vakil, Algebraic structures on the topology of moduli spaces of curves and maps, in Surveys in Differential Geometry Vol. 14: Geometry of Riemann surfaces and their moduli spaces, L. Ji, S. Wolpert, S.-T. Yau eds., Int. Press, Boston, [14] F. Sottile, R. Vakil, and J. Verschelde, Solving Schubert Problems with Littlewood- Richardson Homotopies, in the Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, Stephen W. Watt ed., ACM 2010, p [15] R. Vakil, The moduli space of curves and Gromov-Witten theory, in Enumerative invariants in algebraic geometry and string theory, Lectures Notes in Math. 1947, Springer, Berlin, [16] R. Vakil, The Mathematics of Doodling, Amer. Math. Monthly, to appear. [17] R. Vakil and K. Wickelgren, Universal covering spaces and fundamental groups in algebraic geometry as schemes, submitted for publication. [18] R. Vakil and A. Zinger, A desingularization of the main component of the modlui space of genus-one stable maps to projective space, Geom. and Topol. 12 (2008) no. 1, [19] R. Easton and R. Vakil, Absolute Galois acts faithfully on the components of the moduli space of surfaces: A Belyi-type theorem in higher dimension, Int. Math. Res. Notices IMRN 2007, no. 20, Art. ID rnm080, 10 pp. [20] I. P. Goulden, D. M. Jackson, and R. Vakil, The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals, Proc. L.M.S. (3), 83 (2001), no. 3, [21] I. P. Goulden, D. M. Jackson, and R. Vakil, Towards the geometry of double Hurwitz numbers, Adv. Math. 198 (2005), [22] T. Graber and R. Vakil, Hodge integrals, Hurwitz numbers, and virtual localization, Compositio Math. 135 (1), January 2003, [23] T. Graber and R. Vakil, Relative virtual localization, and vanishing of tautological classes on moduli spaces of curves, Duke Math. J. 30, no. 1, 2005, [24] M. Lieblich, M. Olsson, B. Osserman, and R. Vakil, Deformations and Moduli in Algebraic Geometry, book in preparation, currently 257 pages. [25] M. Roth and R. Vakil, The affine stratification number and the moduli space of curves, in Procedings of the Workshop on algebraic structures and moduli spaces, CRM Proceedings and Lecture Notes, Université de Montréal , [26] R. Vakil, Murphy s Law in algebraic geometry: Badly-behaved deformation spaces, Invent. Math. 164 (2006),

3 [27] R. Vakil, A geometric Littlewood-Richardson rule (with an appendix joint with A. Knutson), Annals of Math. 164 (2006), [28] R. Vakil, Schubert induction, Annals of Math. 164 (2006), [29] R. Vakil and A. Zinger, A natural smooth compactification of the space of eliptic curves in projective space, Elec. Res. Ann. of the Amer. Math. Soc. 13 (2007), [30] R. Vakil, A Mathematical Mosaic: Patterns and Problem-Solving, second expanded edition, Math. Assoc. Amer., [31] R. Vakil, Foundations of Algebraic Geometry, notes in progress, currently 505 pages, current version released through a worldwide online reading course at [32] N. Arkani-Hamed, J. Bourjaily, F. Cachazo, J. Trnka, Unification of residues and Grassmannian dualities, arxiv: [33] N. Arkani-Hamed, J. Bourjaily, F. Cachazo, S. Caron-Huot, J. Trnka, The all-loop integrand for scattering amplitudes in planar N =4SYM, arxiv: [34] N. Arkani-Hamed, F. Cachazo, C. Cheung, and J. Kaplan, A duality for the S matrix, arxiv: v1. [35] N. Arkani-Hamed, F. Cachazo, and C. Cheung, The Grassmannian origin of dual superconformal invariance, arxiv: v1. [36] K. Behrend and A. O Halloran, On the cohomology of stable maps spaces, Invent. Math. 154 (2003) no. 2, [37] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B405 (1993), [38] T. Bridgeland, An introduction to motivic Hall algebras, arxiv: v1. [39] R. Cavalieri, P. Johnson, and H. Markwig, Chamber structure of double Hurwitz numbers, arxiv [40] L. Chen, Y. Li and K. Liu, Localization, Hurwitz numbers, and the Witten conjecture, Asian J. Math. 12 (2009), no. 4, [41] F. Cools, J. Draisma, S. Payne, and E. Robeva, A tropical proof of the Brill-Noether Theorem, arxiv: [42] C. Faber, Chow rings of moduli spaces of curves I: The Chow ring of M 3, Ann. of Math. (2) 132 (1990), no. 2,

4 [43] C. Faber, Chow rings of moduli spaces of curves II: Some results on the Chow ring of M 4, Ann. of Math. (2) 132 (1990), no. 3, [44] C. Faber, A conjectural description of the tautological ring of the moduli space of curves, in Moduli of curves and abelian varieties, , Aspects Math., E33, Vieweg, Braunschweig, [45] E. Getzler and R. Pandharipande, The Betti numbers of M 0,n (r, d), J. Alg. Geom. 15 (2006), no. 4, , math.ag/ [46] T. Graber and R. Pandharipande, Constructions of nontautological classes on moduli spaces of curves, Michigan Math. J. 51 (2003), no. 1, [47] E. Izadi, The Chow ring of the moduli space of curves of genus 5, in The moduli space of curves (Texel Island, 1994), , Progr. Math., 129, Birkhäuser Boston, Boston, MA, [48] P. Johnson, Double hurwitz numbers via the infinite wedge, arxiv: [49] J. Kaplan, Unraveling L(n, k): Grassmannian kinematics, arxiv: (JHEP 1003:025, 2010). [50] M. E. Kazarian, KP hierarchy for Hodge integrals, Adv. Math. 221 (2009), no. 1, [51] L. Krieger, Stanford and UC Berkeley create massively collaborative math..., San Jose Mercury News, August 8, [52] A. Knutson, Puzzles, positroid varieties, and equivariant K-theory of Grassmannians, arxiv: [53] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, [54] K. Liu and H. Xu, A proof of the Faber intersection number conjecture, J. Diff. Geom. 83 (2009), no. 2, [55] D. Mumford, Toward an enumerative geometry of the moduli space of curves, in Arithmetic and geometry, Vol. II, , Prog. Math. 36, Birk. Boston, Boston, MA, [56] R. Pandharipande, Three questions in Gromov-Witten theory, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), , Higher Ed. Press, Beijing, [57] S. Shadrin, On the structure of Goulden-Jackson-Vakil formula, Math. Res. Lett. 16 (2009), no. 4, , arxiv

5 [58] S. Shadrin and D. Zvonkine, Change of variables in ELSV-type formulas, Michigan Math. J. 55 (2007), , arxiv: [59] A. Snowden, Syzygies of Segre embeddings, arxiv: v2, submitted. [60] A. Vistoli, Chow groups of quotient varieties, J. Algebra 107 (1987), [61] E. Witten, Two dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom. 1 (1991), [62] A. Zinger, The reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces, J. Amer. Math. Soc. 22 (2009), no. 3,

On the tautological ring of M g,n

Proceedings of 7 th Gökova Geometry-Topology Conference, pp, 1 7 On the tautological ring of M g,n Tom Graber and Ravi Vakil 1. Introduction The purpose of this note is to prove: Theorem 1.1. R 0 (M g,n