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
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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.

  • 1
    $\begingroup$ 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
  • 5
    $\begingroup$ @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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.