Let $\overline{M}_{g,A}$ the moduli stack of pointed genus $g$ stable curves with weights $A = (a_1,...,a_n)$ introduced in Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352

In this paper the author proves that $\overline{M}_{g,A}$ is indeed a smooth DM-stack endowed with a universal curve $\pi:\mathcal{C}_{g,A}\rightarrow \overline{M}_{g,A}$.

Can we identify $\mathcal{C}_{g,A}$ with another space $\overline{M}_{g,A\cup \{a_{n+1}\}}$ with one more marked point and $\pi$ with the natural morphism forgetting the additional point? Or at least can we do this at the level of the coarse moduli spaces?


There is probably not an isomorphism.

First, note a reason to be suspicious - if you had such an isomorphism, couldn't you just iterate it to get an isomorphism $\overline{M}_{g,A}=\overline{\mathcal M}_{g,n}$. This would make the work of Brendan Hasset in his paper somewhat superfluous.

I claim there is a natural map $ \overline{M}_{g, A \cup \{a_n+1 \} } \to \mathcal C_{g,A} $. To construct it, simply delete the last marked point. If the twisted canonical bundle ceases to be ample on the component the last marked point was on, contract it. Repeat until you have a weighted stable curve with $n$ marked points. Then choose from among the points of this curve the image of the last marked point under the contraction map.

However, this map often fails to be injective. Suppose there is some $a_i, a_j$ with $a_i+a_j\leq 1$, $a_i + a_j + a_{n+1} >1$. Consider a genus $g$ smooth curve glued to a single copy of $\mathbb P^1$ where points $i,j,n+1$ are on the $\mathbb P^1$ and the other points are distributed on the genus $g$ curve. Removing point $a_{n+1}$ causes the $\mathbb P^1$ to be contracted, and $a_i$ and $a_j$ become the same point. This happens regardless of the location of $a_i, a_j, a_{n+1}$ on the old $\mathbb P^1$. But there is a 1-parameter family of possible locations of these three points and the one node up to isomorphism. This curve is contracted to a point by the map $ \overline{M}_{g, A \cup \{a_n+1 \} } \to \mathcal C_{g,A} $.

Because the other points can be chosen on the genus $g$ curve to kill all the automorphisms, passing to the coarse moduli space does not change anything in this example.

  • $\begingroup$ Thanks a lot for the answers. However I completely agree with you in the case $A = (1,...,1)$ i.e $\overline{M}_{g,A}\cong\overline{M}_{g,n}$ but I am not sure that this process will work with general weights. What do you think? $\endgroup$ – JdiNirv Apr 12 '15 at 15:41
  • $\begingroup$ @JdiNirv Fixed. $\endgroup$ – Will Sawin Apr 12 '15 at 17:02
  • $\begingroup$ I see. Thanks. Anyway do you think it is possible to construct a section of a forgetful morphism $\overline{M}_{g,A[n+1]}\rightarrow\overline{M}_{g,A[n]}$ with $A[n] = (a_1,...,a_n)$ and $A[n+1] = (a_1,...,a_n,a_{n+1})$? $\endgroup$ – JdiNirv Apr 12 '15 at 17:13
  • $\begingroup$ @JdiNirv The only method I can think of to construct a section is to use one of the marked points and put another marking there, blowing it up if the sums of the markings exceeds $1$. I think this will not always be well-defined but it is in some cases - for instance in the case where all $a_i$ are $1$. $\endgroup$ – Will Sawin Aug 9 '15 at 3:11

As Hassett notes in Section 2.1.1, the universal curve $\mathcal{C}_{g,A}$ can be identified with the slightly generalized Hassett moduli space $\overline{M}_{g,A\cup \{0\}}$, where weight zero markings are allowed to lie on singular points. This is clear since the moduli problem is identical.

As Will Sawin notes (and also can be found in Remark 4.2 in Hassett's paper) there is a birational reduction map $\overline{M}_{g,A\cup \{a_{n + 1}\}} \to \overline{M}_{g,A\cup \{0\}}$.

The space of weights $(a_1, \dotsc, a_n)$ has a wall and chamber structure and the moduli spaces of weighted stable curves do not change within each chamber. In our case, this implies that the map $\overline{M}_{g,A\cup \{a_{n + 1}\}} \to \overline{M}_{g,A\cup \{0\}}$ is an isomorphism for $a_{n + 1}$ small enough, assuming that no subset $S$ of elements of $A$ such that $2 \le |S| \le n - 2$ has $\sum_{a \in S} a = 1$.Therefore, under this assumption on the weights, we can identify $\mathcal{C}_{g,A}$ with a usual moduli space of weighted stable curves. This is Proposition 5.4 in Hassett's paper.


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