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?)



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


