# 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 ).

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 , where $M_2$ is explicitly described).

A general method for proving uni-rationality of $M_g$ has been constructed . 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, ): 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).

How to Cite This Entry:
Moduli problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moduli_problem&oldid=32397
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article