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I am reading moduli spaces of $n$-pointed rational stable curves denoted by $\overline{M_{0,n}}$. I am not understanding intersection of some divisors as varieties. We know there are forgetful morphisms $\pi_{i,j,k,l}:\overline{M_{0,n}}\to\overline{M_{0,4}}=\mathbb{P}^1$. Let $x\in M_{0,n}$ and we denote $D_{i,j,k,l}=\pi_{i,j,k,l}^{-1}\{\pi_{i,j,k,l}(x)\}$ my questions are following:

  1. Does $D_{i,j,k,l}$ intersects $D_{i',j',k',l'}$ transversally for any {i,j,k,l,i',j',k',l'}$\subset${$1,\cdots,n$}?

  2. What is the dimension of $D_{i,j,k,l}\cap D_{i',j',k',l'} $?

  3. can $\cap D_{i,j,k,l}\subset\overline{M_{0,n}}- M_{0,n}$ for some {i,j,k,l}$\subset${1,$\cdots$,n}?

Any comments or reference will be very helpful. If someone can comment for some specific example say for $n=7$ that is also appreciated.

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    $\begingroup$ The answers to the first question is "yes" whenever $\{i,j,k,\ell\}$ is different from $\{i',j',k',\ell'\}$, and then the intersection has dimension $\text{dim}(\overline{M}_{0,n})-2$. The answer to the third question is "no", precisely because you posit that each $D_{i,j,k,\ell}$ contains the specified point $x$, and $x$ is an element in $M_{0,n}$. $\endgroup$
    – Jason Starr
    Commented Jun 23, 2023 at 11:11
  • $\begingroup$ Thanks for your comment @JasonStarr. Is it true that $D_{i,j,k,l}$ on $M_{0,n}$ is like projection from the product and so $\cap D_{i,j,k,l}-x$ is contained in boundary for some ${i,j,k,l}$? $\endgroup$
    – gary
    Commented Jun 23, 2023 at 13:07
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    $\begingroup$ I am not sure precisely what you mean. If you fix an $n$-pointed, projective, geometrically connected, smooth curve $(C,p_1,\dots,p_n)$, then the natural $1$-morphism from $C^n\setminus \Delta$ to $M_{0,n}$ is smooth and (geometrically) surjective. This map is "compatible" with projections to $C^m\setminus \Delta$ for $m\leq n$. Is that what you are asking about? $\endgroup$
    – Jason Starr
    Commented Jun 23, 2023 at 14:53
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    $\begingroup$ If I properly understand your updated second question: yes, for every three-element subset $\{i,j,k\}$ of $\{1,\dots,n\}$, the intersection over all $\ell\in \{1,\dots,n\} \setminus \{i,j,k\}$ of $D_{i,j,k,\ell}$ is precisely the singleton subset $\{x\}$ of the moduli point of $(C,p_1,\dots,p_n)$. There are no other elements in this intersection: no other elements in $M_{0,n}$, nor any other elements in $\overline{M}_{0,n}$. $\endgroup$
    – Jason Starr
    Commented Jun 24, 2023 at 11:06
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    $\begingroup$ Every intersection point parameterizes a stable, marked curve that has the same $\{i,j,k,\ell\}$-cross ratios as $(C,p_1,\dots,p_n)$, and these are non-degenerate cross ratios by hypothesis. So the stable, marked curve is a point in the “interior” $\M_{0,n}$. $\endgroup$
    – Jason Starr
    Commented Jun 24, 2023 at 18:50

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