Pieter Belmans took the table to the 21st century making it interactive.
In the future, we intend to make it a repository for M_{g,n} knowledge and open questions. If you have any suggestions please let us know!
Explanation
This is a picture that summarizes the state of the art on the Kodaira classification of the moduli spaces M_{g,n}. Blue means Kodaira dimension negative, red means general type, and yellow intermediate type or partial results. The first version of this table was made by Gregor Botero.
If there are inaccuracies or updates to be made please let me know!
References in historical order.
For n=0:
F. Severi, Sulla classificazione delle curve algebriche e sul teorema d’esistenza di Riemann, Rendiconti della R. Accad. Naz. Lincei 24 (1915), 877-888.
J. Harris and D. Mumford, On the Kodaira dimension of M_g, Invent. Math. 67 (1982), 23-88.
M. C. Chang and Z. Ran, Unirationality of the moduli space of curves of genus 11, 13 (and 12), Invent. Math. 76 (1984), 41-54.
M. C. Chang and Z. Ran, The Kodaira dimension of the moduli space of curves of genus 15, J. Differential Geom. 24 (1986), 205-220.
D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus ≥ 23, Invent. Math. 90 (1987), 359-387.
A. Bruno and A. Verra, M_15 is rationally connected, in: Projective Varieties with Unexpected Properties, (Walter de Gruyter GmbH, Berlin, 2005), 51-65.
G. Farkas, D. Jensen, and S. Payne, The Kodaira dimensions of M_22 and M_23, Preprint, 2020, arXiv: 2005.00622.
G. Farkas and A. Verra, On the Kodaira dimension of M_16, Preprint, 2020, arXiv: 2008.08852.
D. Agostini and I. Barros, Pencils on surfaces with normal crossings and the Kodaira dimension of M_{g,n}, Forum Math. Sigma 9 (2021), E31, 1-22.
For n>0:
P. Belorousski, Chow rings of moduli spaces of pointed elliptic curves, Ph.D. thesis, University of Chicago, Chicago, 1998.
A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125(1) (2003), 105-138.
G. Farkas and M. Popa, Effective divisors on M_g, curves on K3 surfaces and the Slope Conjecture, J. Algebraic Geom. 14 (2005), 151-174.
G. Bini and C. Fontanari, Moduli of curves and spin structures via algebraic geometry, Trans. Amer. Math. Soc. 358 (2006), 3207-3217.
G. Farkas, Koszul divisors on moduli spaces of curves. Amer. J. Math. 131 (2009), 819-867.
G. Farkas and A. Verra, The classification of universal Jacobians over the moduli space of curves, Comment. Math. Helv. 88 (2013), 587-611.
L. Benzo, Uniruledness of some moduli spaces of stable pointed curves, J. Pure Appl. Algebra 218(3) (2014), 395-404.
I. Kadiköylü, Maximal rank divisors on M_{g,n}, Ann. Sc. Norm. Super. Pisa Cl. Sci. 20(5) (2020), no. 1, 349-371.
I. Barros and S. Mullane, Two moduli spaces of Calabi-Yau type, Int. Math. Res. Not. IMRN 2021 (2021), 15833-15849.
D. Agostini and I. Barros, Pencils on surfaces with normal crossings and the Kodaira dimension of M_{g,n}, Forum Math. Sigma 9 (2021), E31, 1-22.
I. Barros and S. Mullane, The Kodaira classification of the moduli of hyperelliptic curves, Preprint, 2022, arXiv: 2106.13774.
G. Farkas, D. Jensen, and S. Payne, The non-abelian Brill-Noether divisor on M_{13} and the Kodaira dimension of R_{13}, Preprint, 2021, arXiv: 2110.09553.