Moduli problem

The classical problem of the rationality or uni-rationality of the moduli variety of algebraic curves of genus $g$.

Riemann surfaces of genus $g$ (up to isomorphism) depend on $3g-3$ complex parameters — the moduli (see Moduli of a Riemann surface). The set of classes of non-singular projective curves of genus $g$ over an algebraically closed field $k$ has the structure of a quasi-projective algebraic variety $M_g$ (see [3]–[5]).

The manifolds $M_g$ for $g=0$ and 1 have a simple structure: $M_0$ consists of one point, and $M_1$ is isomorphic to the affine line $A^1$. Therefore the moduli problem refers to curves of genus $g\geq2$ and is formulated as follows: Is the moduli variety $M_g$ of curves of genus $g\geq2$ rational, or at least uni-rational? The rationality of $M_g$ has been established only for $g=2$ (see [2], where $M_2$ is explicitly described).

A general method for proving uni-rationality of $M_g$ has been constructed [6]. By this method, in particular, the uni-rationality of $M_g$ for all $g\leq10$ has been proved. The uni-rationality of $M_{11}$, $M_{12}$ and $M_{13}$ has also been proved.

The moduli problem frequently receives a broader interpretation (see, for example, [5]): It refers to the whole complex of problems associated with the existence of moduli spaces of certain algebraic objects (varieties, vector bundles, endomorphisms, etc.), with the study of their various algebraic-geometric properties and with compactification techniques for moduli spaces (see Moduli theory).

References

[1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
[2]	J. Igusa, "Arithmetic variety of moduli for genus two" Ann. of Math. , 72 : 3 (1960) pp. 612–649 MR0114819 Zbl 0122.39002
[3]	D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[4]	D. Mumford, "Stability of projective varieties" l'Enseign. Math. (2) , 23 : 1–2 (1977) pp. 39–110 MR0450273 MR0450272 Zbl 0497.14004 Zbl 0376.14007 Zbl 0363.14003
[5]	H. Popp, "Moduli theory and classification theory of algebraic varieties" , Springer (1977) MR0466143 Zbl 0359.14005
[6]	F. Severi, "Sulla classificazione delle curve algebriche e sul teorema d'esistenza di Riemann" Atti R. Accad. Naz. Lincei Rend. , 24 (1915) pp. 877–888

Comments

It is now known that $M_g$ is of general type (cf. General-type algebraic surface) for $g\geq24$ and has positive Kodaira dimension for $g=23$ (cf. [a1], [a2]); thus $M_g$ is not uni-rational for $g\geq23$. For $g=11,12,13$, $M_g$ is uni-rational. Also, $M_{15}$ has negative Kodaira dimension. The nature of $M_g$ for $g=16,\ldots,22$ is still (1989) unknown.

References

[a1]	M. Chang, Z. Ran, "Unirationality of the moduli space of curves of genus 11, 13 (and 12)" Invent. Math. , 76 (1984) pp. 41–54
[a2]	D. Eisebud, J. Harris, "The Kodaira dimension of the moduli space of curves of genus $g\geq23$" Invent. Math. , 90 (1987) pp. 359–387
[a3]	J. Harris, D. Mumford, "On the Kodaira dimension of the moduli space of curves" Invent. Math. , 67 (1982) pp. 23–86 MR0664324 Zbl 0506.14016
[a4]	S. Mori, S. Mukai, "Uniruledness of the moduli space of curves of genus 11" M. Reynard (ed.) T. Shioda (ed.) , Algebraic geometry , Lect. notes in math. , 1016 , Springer (1983) pp. 334–353 MR0726433 Zbl 0557.14015
[a5]	E. Sernesi, "L'unirazionalità della varietà dei moduli delle curvi di genere dodici" Ann. Scuola Norm. Sup. Pisa (IV) , VIII (1981) pp. 405–439
[a6]	N. Shepherd-Barron, "The rationality of certain spaces associated to trigonal curves" S.J. Bloch (ed.) , Algebraic geometry , Proc. Symp. Pure Math. , 46.1 , Amer. Math. Soc. (1987) pp. 165–171 Zbl 0669.14015
[a7]	N. Shepherd-Barron, "Invariant theory for $S_5$ and the rationality of $M_6$" Compos. Math. , 70 (1989) pp. 13–25 Zbl 0704.14044
[a8]	J. Harris, "Curves and their moduli" S.J. Bloch (ed.) , Algebraic geometry , Proc. Symp. Pure Math. , 46.1 , Amer. Math. Soc. (1985) pp. 99–143 MR0927953 Zbl 0646.14019
[a9]	D. Eisenbud, J. Harris, "Limit linear series" Bull. Amer. Math. Soc. , 10 (1984) pp. 277–280 MR0733695 Zbl 0533.14013