Moduli problem

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

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

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

A general method for proving uni-rationality of has been constructed [6]. By this method, in particular, the uni-rationality of for all has been proved. The uni-rationality of , and 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)
[2]	J. Igusa, "Arithmetic variety of moduli for genus two" Ann. of Math. , 72 : 3 (1960) pp. 612–649
[3]	D. Mumford, "Geometric invariant theory" , Springer (1965)
[4]	D. Mumford, "Stability of projective varieties" l'Enseign. Math. (2) , 23 : 1–2 (1977) pp. 39–110
[5]	H. Popp, "Moduli theory and classification theory of algebraic varieties" , Springer (1977)
[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 is of general type (cf. General-type algebraic surface) for and has positive Kodaira dimension for (cf. [a1], [a2]); thus is not uni-rational for . For , is uni-rational. Also, has negative Kodaira dimension. The nature of for 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 " 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
[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
[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
[a7]	N. Shepherd-Barron, "Invariant theory for and the rationality of " Compos. Math. , 70 (1989) pp. 13–25
[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
[a9]	D. Eisenbud, J. Harris, "Limit linear series" Bull. Amer. Math. Soc. , 10 (1984) pp. 277–280