Consider the (Deligne-Mumford compactification of the) moduli space of complex rational marked curves $\overline M_{0;n}$. For each $i\in \{1,\ldots,n\}$ we can construct a line bundle $L_i$ with a fiber given by cotangent space to the $i$-th marked point, and the divisor corresponding to $L_i$ is the $\psi$-class.

Studying the tropical counterpart of this moduli space, I have managed to compute the dimension of linear system of $\psi_i^{trop}$. Namely

$L (\psi_i^{trop}) = 2{n-1\choose 2} - 4.$

I was wondering, does this formula holds in the complex case as well? I.e. what is the dimension of the linear system of the $\psi$-class for the moduli space of complex rational marked curves?

One can see, that the result agrees in the case $n=4$. The $\psi$-class for $n=4$ is just a degree 1 effective divisor, and Riemann-Roch formula gives $L(\psi_i) =2$.

Thanks in advance.


I think that this is done by Kapranov in "Veronese curves and Grothendiexk-Knudsen moduli space $M_{0,n}$". The projective dimension of this linear system should always be $n-3$ (so the linear dim is $n-2$) and the map is a birational morphism which is the inverse of a sequence of blow-ups.

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