Timeline for On the algberaicity of the universal elliptic curve associated to a torsion free subgroup
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 31, 2012 at 21:00 | vote | accept | Hugo Chapdelaine | ||
| May 31, 2012 at 21:00 | history | bounty ended | Hugo Chapdelaine | ||
| May 30, 2012 at 19:15 | comment | added | Hugo Chapdelaine | Yes of course you are right! | |
| May 30, 2012 at 17:03 | comment | added | Misha | @Hugo: Of course, there will be non-congruence finite covers, for them you argue as follows. Every non-congruence finite cover $C_1$ and a congruence-cover $C_2$ (of the original complex space/variety) have a common finite regular cover $C_3\to C_1$, $C_3\to C_2$. Since you know that $C_2$ is quasi-projective, it follows that $C_3$ is also quasi-projective (pull-back of positive bundle is positive). But finite quotient $C_1$ of the quasi-projective variety $C_3$ is then quasi-projective as well. | |
| May 30, 2012 at 10:38 | comment | added | Hugo Chapdelaine | @Misha, so what you told me in the comments look good, I like this idea of using an explicit positive line bundle. So concerning your addendum keep in mind that not all finite index subgroups are congruence subgroups. So I don't quite understand why your $X_d(N)$ is projective... | |
| May 30, 2012 at 4:42 | comment | added | Misha | Pull-back bundle $f^*(L)$ on $\bar{M}^o_{g,n}$ is still positive, so $\bar{M}^o_{g,n}$ is projective (by Kodaira) provided that it is a complex manifold. The latter you get by taking $Mod^o_{g,n}$ to be any finite-index subgroup in an appropriate congruence-subgroup in $Mod_{g,n}$ (as in Looijenga's paper on Prym level structures staff.science.uu.nl/~looij101/prymlevel2.ps). Projectivity of the rest of quotients of $T_{g,n}$ by finite-index subgroups in $Mod_{g,n}$ follows by taking finite quotients of $M^o_{g,n}$'s above. | |
| May 30, 2012 at 4:25 | comment | added | Misha | @Hugo: As an alternative to differential-geometric arguments you can use the following: The DM compactification $\bar{M}_{g,n}$ of the moduli space $M_{g,n}=T_{g,n}/Mod_{g,n}$ is projective. Hence, it admits a positive line bundle $L$ (here I regard $\bar{M}_{g,n}$ as a complex space rather than a stack). For every finite-index subgroup $Mod^o_{g,n}\subset Mod_{g,n}$, the quotient $M^o_{g,n}=T_{g,n}/Mod^o_{g,n}$ admits a compactification to a complex space $\bar{M}^o_{g,n}$ and a holomorphic branched cover $f: \bar{M}^o_{g,n}\to \bar{M}_{g,n}$. ... | |
| May 29, 2012 at 22:01 | history | edited | Misha | CC BY-SA 3.0 |
Added reference to Livne's thesis
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| May 29, 2012 at 21:42 | comment | added | Misha | One can use S.Wolpert "On obtaining a positive line bundle from the Weil-Petersson class," Amer. J. Math. 107 (1985), no. 6, 1485–1507 (1986). In this paper Wolpert proves that the DM compactification of the moduli space is projective by a local computation, proving positivity of current defined by Weil-Peterson metric, and, hence, positivity of the corresponding line bundle. Since the computation is local, it goes through for finite-index subgroups of the mapping class group. I am also adding an alternative reference for the main result in my main answer. | |
| May 29, 2012 at 20:04 | comment | added | Hugo Chapdelaine | @Misha, do you have a nice reference (which covers the case that I'm interested in) which shows in details why is $T(S)/Mod_S^{\circ}$ algebraic. I would like to see how one uses torsion-freeness in order to deduce algebraicity, which is the whole point in my question. | |
| May 29, 2012 at 18:22 | history | answered | Misha | CC BY-SA 3.0 |