The following question concerns Deligne-Mumford compactification of the (coarse) moduli space $M_{g,n}$ of smooth complex genus g curves with $n$ marked points.

If there are no marked points (ie $n=0$), then given a family of curves $C_\lambda$ given by $w^N=z^a (z-1)^b (z-\lambda)^c,$ as $\lambda\to 0$ I believe the limit is the stable curve $w^N=z^{a+c} (z-1)^b.$

However, if all lifts of $0, 1, \lambda, \infty$ are marked points, I no longer know how to determine the limit as $\lambda\to0$ in $\bar{M_{g,n}}$.

So, my question is, how do you do such explicit computations when there are marked points?

  • $\begingroup$ If $N>2$ your original curves are not stable unless $a=b=c=1$ and then the limit curve is not stable unless $a+b\leq1$ (if $N=2$ the conditions are a little bit less stringent). $\endgroup$ – Torsten Ekedahl Nov 22 '10 at 5:20
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    $\begingroup$ About the original curves: I mean the Riemann surfaces obtained as the normalization of the algebraic curve with given equation. $\endgroup$ – Alex Nov 22 '10 at 5:29
  • $\begingroup$ In that case your problem doesn't always seem well defined, when $N$ is not relatively prime to one of $a$, $b$, $c$ or $a+b+c$ there will be several points above one of $0$, $1$, $\lambda$ or $\infty$ and you have to tell which one you mark. $\endgroup$ – Torsten Ekedahl Nov 22 '10 at 7:29
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    $\begingroup$ I wish to mark all points above 0, 1, $\lambda$ and $\infty$. $\endgroup$ – Alex Nov 22 '10 at 16:42
  • $\begingroup$ Perhaps the first step is for you to describe the normalized family in equations. $\endgroup$ – S. Carnahan Dec 15 '10 at 8:11

Marked points do not collide even on $\overline{M}_{g,n}$. So, your original family will no longer be stable: the limit curve will need to have an extra component.

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    $\begingroup$ Could you provide more details? For example, how do you determine what the components look like and how many components there are? $\endgroup$ – Alex Nov 22 '10 at 5:16
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    $\begingroup$ @Alex : I recommend learning what is going on first in the case $g=0$, where things are easier. For this, I recommend reading Chapter 1 of Kock-Vainsencher's "An invitation to quantum cohomology". $\endgroup$ – Andy Putman Nov 22 '10 at 6:40
  • $\begingroup$ @Andy: Could you provide a sketch of the ideas here? $\endgroup$ – Dr Shello Dec 15 '10 at 8:40
  • $\begingroup$ Even better start with $g=0$ and $n=4$ so your moduli space is $\mathbb{P}^1$, given by the cross-ratio $\lambda$ of those points, so we may as well fix the first 3 in order as $0,1,\infty$. Once we do that there are no automorphisms of this pointed curve. If $\lambda$ collides with one of them, we have automorphisms so we have to blow up that (unstable) point. We are left then with 2 $\mathbb{P}^1$'s with 2 marked points meeting at a distinct point. $$ $$ For larger $n$ you just perform that sort of operation many times. For larger $g$ you might have automorphisms of the pointed curve. $\endgroup$ – stankewicz Dec 15 '10 at 15:10

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