View source for Moduli problem

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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|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 [[#References|[3]]]–[[#References|[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 [[#References|[2]]], where $M_2$ is explicitly described).

A general method for proving uni-rationality of $M_g$ has been constructed [[#References|[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, [[#References|[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|Moduli theory]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Igusa, "Arithmetic variety of moduli for genus two" ''Ann. of Math.'' , '''72''' : 3 (1960) pp. 612–649 {{MR|0114819}} {{ZBL|0122.39002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Geometric invariant theory" , Springer (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Mumford, "Stability of projective varieties" ''l'Enseign. Math. (2)'' , '''23''' : 1–2 (1977) pp. 39–110 {{MR|0450273}} {{MR|0450272}} {{ZBL|0497.14004}} {{ZBL|0376.14007}} {{ZBL|0363.14003}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H. Popp, "Moduli theory and classification theory of algebraic varieties" , Springer (1977) {{MR|0466143}} {{ZBL|0359.14005}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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</TD></TR></table>



====Comments====
It is now known that $M_g$ is of general type (cf. [[General-type algebraic surface|General-type algebraic surface]]) for $g\geq24$ and has positive [[Kodaira dimension|Kodaira dimension]] for $g=23$ (cf. [[#References|[a1]]], [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Chang, Z. Ran, "Unirationality of the moduli space of curves of genus 11, 13 (and 12)" ''Invent. Math.'' , '''76''' (1984) pp. 41–54</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D. Eisebud, J. Harris, "The Kodaira dimension of the moduli space of curves of genus $g\geq23$" ''Invent. Math.'' , '''90''' (1987) pp. 359–387</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Harris, D. Mumford, "On the Kodaira dimension of the moduli space of curves" ''Invent. Math.'' , '''67''' (1982) pp. 23–86 {{MR|0664324}} {{ZBL|0506.14016}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> 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 {{MR|0726433}} {{ZBL|0557.14015}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E. Sernesi, "L'unirazionalità della varietà dei moduli delle curvi di genere dodici" ''Ann. Scuola Norm. Sup. Pisa (IV)'' , '''VIII''' (1981) pp. 405–439</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|0669.14015}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> N. Shepherd-Barron, "Invariant theory for $S_5$ and the rationality of $M_6$" ''Compos. Math.'' , '''70''' (1989) pp. 13–25 {{MR|}} {{ZBL|0704.14044}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> 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 {{MR|0927953}} {{ZBL|0646.14019}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> D. Eisenbud, J. Harris, "Limit linear series" ''Bull. Amer. Math. Soc.'' , '''10''' (1984) pp. 277–280 {{MR|0733695}} {{ZBL|0533.14013}} </TD></TR></table>