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It is known that the moduli space $\overline{M}_{g}$ of genus $g$ curves is of general type for $g\geq 24$.

By Theorem 2.4 of

Logan, Adam The Kodaira dimension of moduli spaces of curves with marked points. Amer. J. Math. 125 (2003), no. 1, 105–138.

all but finitely many of the $\overline{M}_{g,n}$ with $g>3$ are of general type. Furthermore in Theorem 5.1 of the same paper for any $4\leq g\leq 23$ an $n$ such that $\overline{M}_{g,n}$ is of general type is determined.

Is there an inequality in terms of $g,n$ which implies that $\overline{M}_{g,n}$ is of general type ?

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  • $\begingroup$ I am not sure what you are asking. In any case you must impose $g\geq 4$ since the result is not known for $\bar{M}_{2,n}$ and $\bar{M}_{3,n}$. Then you may observe that Logan's inequalities imply (trivially) that $\bar{M}_{g,n}$ is of general type for $g+n\geq 30$. Does that answer your question? $\endgroup$
    – abx
    Commented Nov 3, 2013 at 15:02
  • $\begingroup$ How do you see it for instance for $g=22, n=9$ and for $g=23, n = 7$ ? $\endgroup$
    – Puzzled
    Commented Nov 3, 2013 at 15:16
  • $\begingroup$ According to Logan, $n=8$ is enough for $g=22$, and $n=1$ for $g=23$. $\endgroup$
    – abx
    Commented Nov 3, 2013 at 16:23
  • $\begingroup$ Thank you very much. Do you have a reference for the statement: general type on the base + general type on the general fiber = the variety is of general type ? $\endgroup$
    – Puzzled
    Commented Nov 3, 2013 at 16:54
  • $\begingroup$ I read the comment down the page. Thanks again. $\endgroup$
    – Puzzled
    Commented Nov 3, 2013 at 16:55

1 Answer 1

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So it seems your question is: if we know that $\overline M_{g,n}$ is of general type, is the same true for $\overline M_{g,n'}$ with $n' \geq n$? The answer is yes, and is a special case of the main theorem of: J. Kollár, Subadditivity of the Kodaira dimension : fibers of general type.

As far as I know there are no explicit counterexamples to the stronger statement that if $\overline M_{g,n}$ is of general type, then the same true for $\overline M_{g',n}$ with $g' \geq g$ and $n' \geq n$, but it is almost certainly false: for instance, Farkas has proved that $\overline M_{22}$ is of general type and also that $\kappa(\overline M_{23}) \geq 2$, which he however conjectured to be sharp. Also note that Logan's function $f(g)$ such that $\overline M_{g,n}$ is of general type for $n \geq f(g)$ is not monotone in $g$.

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  • $\begingroup$ Thank you. I see but the function $f(g)$ is just on finitely many $g$. Furthermore for any such $g$ he find an $n$ such that $\overline{M}_{g,n}$ is of general type but he does not state it for any $n\geq f(g)$. $\endgroup$
    – Puzzled
    Commented Nov 3, 2013 at 16:13
  • $\begingroup$ I guess that now my question becomes this: let $f:X\rightarrow Y$ be a fibration where $Y$ and the general fiber of $f$ are of general type. Is $X$ of general type? $\endgroup$
    – Puzzled
    Commented Nov 3, 2013 at 16:24
  • $\begingroup$ Yes. So if $\bar{M}_{g,n}$ is of general type, $\bar{M}_{g,n'}$ for $n'\geq n$ is of general type: it maps onto $\bar{M}_{g,n}$ with fibers of general type. $\endgroup$
    – abx
    Commented Nov 3, 2013 at 16:26
  • $\begingroup$ So this more general fact is true: let $f:X\rightarrow Y$ be a fibration where $Y$ and the general fiber of $f$ are of general type. Then $X$ of general type. Do you have a reference for this? $\endgroup$
    – Puzzled
    Commented Nov 3, 2013 at 16:28
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    $\begingroup$ I think this is due to Viehweg : Die Additivität der Kodaira Dimension für projektive Faserräume über Varietäten des allgemeinen Typs. J. Reine Angew. Math. 330 (1982), 132-142. $\endgroup$
    – abx
    Commented Nov 3, 2013 at 16:54

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