Namespaces
Variants
Actions

Difference between revisions of "Moduli problem"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
(TeX)
 
Line 1: Line 1:
The classical problem of the rationality or uni-rationality of the moduli variety of algebraic curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m0645001.png" />.
+
{{TEX|done}}
 +
The classical problem of the rationality or uni-rationality of the moduli variety of algebraic curves of genus $g$.
  
Riemann surfaces of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m0645002.png" /> (up to isomorphism) depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m0645003.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m0645004.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m0645005.png" /> has the structure of a quasi-projective algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m0645006.png" /> (see [[#References|[3]]]–[[#References|[5]]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m0645007.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m0645008.png" /> and 1 have a simple structure: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m0645009.png" /> consists of one point, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450010.png" /> is isomorphic to the affine line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450011.png" />. Therefore the moduli problem refers to curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450012.png" /> and is formulated as follows: Is the moduli variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450013.png" /> of curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450014.png" /> rational, or at least uni-rational? The rationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450015.png" /> has been established only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450016.png" /> (see [[#References|[2]]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450017.png" /> is explicitly described).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450018.png" /> has been constructed [[#References|[6]]]. By this method, in particular, the uni-rationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450020.png" /> has been proved. The uni-rationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450023.png" /> has also been proved.
+
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]]).
 
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]]).
Line 15: Line 16:
  
 
====Comments====
 
====Comments====
It is now known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450024.png" /> is of general type (cf. [[General-type algebraic surface|General-type algebraic surface]]) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450025.png" /> and has positive [[Kodaira dimension|Kodaira dimension]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450026.png" /> (cf. [[#References|[a1]]], [[#References|[a2]]]); thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450027.png" /> is not uni-rational for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450028.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450030.png" /> is uni-rational. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450031.png" /> has negative Kodaira dimension. The nature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450033.png" /> is still (1989) unknown.
+
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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450034.png" />" ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450035.png" /> and the rationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064500/m06450036.png" />" ''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>
+
<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>

Latest revision as of 22:00, 7 July 2014

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
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