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A famous conjecture of Looijenga states that the moduli space of curves $M_{g,n}$ is the union of $g- \delta_{0,n}+ \delta_{0,g}$ open affine subsets, where $g,n$ are non-negative integers satisfying $2g-2+n>0$, and $\delta$ is the Kronecker delta.

I know of proofs of this conjecture in the case $(g,0)$ for $2 \leq g \leq 5$ (Fontanari-Looijenga and Fontanari-Pascolutti), and in the case $(0,n)$ for all $n$'s.

Is this conjecture true for $M_{1,n}$ ? Namely, is it known if $M_{1,n}$ is affine?

(Are there other cases when the conjecture is known?)

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Yes, $M_{1,n}$ is affine. More generally, $M_{g,n+1}\to M_{g,n}$ is an affine morphism for any $n \geq 1$.

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  • $\begingroup$ Thank you very much! How do you see that the morphism is affine? $\endgroup$
    – OldMacdonaldHadaForm
    Commented Jul 17, 2012 at 11:51
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    $\begingroup$ Let $C \to M_{g,n}$ be the universal curve. The union of the $n$ markings on $C$ is a relative Cartier divisor which is relatively ample (this can be checked on closed points). So its complement is affine. $\endgroup$
    – Dan Petersen
    Commented Jul 17, 2012 at 13:02
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    $\begingroup$ @OMHF -- Although it is true that there is no universal curve over $M_{g,n}$ for every choice of $g$ and $n$, you can use Chevalley's theorem to fix this. There is a finite flat morphism $B\to M_{1,1}$ such that $B$ is an affine scheme and there is a family over $B$, e.g., the $\lambda$-line. By Dan's argument, the associated (2-)fibered product $B\times_{M_{1,1}} M_{1,n}$ is an affine scheme. Since $B \to M_{1,1}$ is finite and surjective, also `$B\times_{M_{1,1}}M_{1,n} \to M_{1,n}$ is finite and surjective. Thus $M_{1,n}$ (as a coarse space) is an affine scheme by Chevalley's thm. $\endgroup$
    – Jason Starr
    Commented Jul 17, 2012 at 20:01
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    $\begingroup$ More elementary is to use that $M_{1,1} = [Y(N)/\mathrm{SL}(2,\mathbf Z/N\mathbf Z)]$ for any $N$, where $Y(N)$ is the (open) modular curve parametrizing elliptic curves with a full level $N$ structure. Now for $N\geq 3$ this is a fine moduli scheme with a universal family; iterating the argument in my previous comment shows that the moduli space of $n$-pointed genus one curves with level $N$ structure is an affine scheme. Dividing by the action of $\mathrm{SL}(2,\mathbf Z/N\mathbf Z)$ recovers $M_{1,n}$. But the quotient of an affine scheme by a finite group is again affine (elementary). $\endgroup$
    – Dan Petersen
    Commented Jul 17, 2012 at 20:46
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    $\begingroup$ (and crucially, $Y(N)$ is affine for any $N$, being a curve minus a finite set of closed points.) $\endgroup$
    – Dan Petersen
    Commented Jul 17, 2012 at 20:54

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