Intersection complex of genus-zero curves?

Question

I would like to have a very explicit description of $\bar M_{0, n}$, especially its boundary divisors and how they intersect. All I can do in my construction is add divisors and blow up at strata, where ``strata'' are the intersections of divisors.

The related ``Losev--Manin'' space $L_n$ is a toric variety, with intersection complex the barycentric subdivision of the boundary of the simplex, otherwise known as the permutohedron.

Constructions I'm aware of:

1. Both $\bar M_{0, n}$ and $L_n$ have ``Kapranov constructions,'' starting with $\mathbb P^{n-3}$ with a certain number of chosen points, blowing up the points, then the lines between them, then the planes between triples of points, etc. To get $L_n$ you start with $n-2$ points; to get $\bar M_{0, n}$ you start with $n-1$.
2. Keel has a construction of $\bar M_{0, n}$ as an iterated blowup of $\bar M_{0, n-1} \times \mathbb P^1$ at certain disjoint closed subschemes.
3. ? Can you describe $\bar M_{0, n+1}$ as the blowup of $\bar M_{0, n} \times_{\bar M_{0, n-1}} \bar M_{0, n}$ along the diagonal? My collaborator says so, but I'm skeptical.

Questions:

1. Can I express the map $\bar M_{0, n} \to L_n$ as first adding divisors to $L_n$ and then blowing up at strata? I want to basically add in the one more point and all its hyperplanes with the other points and then hope that I'm only blowing up at intersections of these divisors. I'm worried the order of blowups matters.

2. If I view $\mathbb P^1$ as $\bar M_{0, 4}$, it has the divisor of three points. Are the loci Keel blows up strata? I.e., are they intersections of the normal-crossing boundary components at each step? I couldn't tell from his paper. I should still have to add divisors first; which are they?

3. Is this construction right, and if so, what divisors do you have to add?

4. A map $\bar M_{0, n} \to L_n$ is given by ``forgetting one of the $n-1$ points'' in the Kapranov construction. So there should be $n-1$ such maps (in fact there are more). Does this give a cover of the intersection complex of $\bar M_{0, n}$ by copies of that of $L_n$, the permutohedron? How explicit can this be made?

Example:

Take $n = 5$. Then $\bar M_{0, n}$ has intersection graph the Petersen graph, while $L_n$ is a hexagon.

Motivation:

If I had a factorization of the map $\bar M_{0, n} \to X_n \to L_n$, where $X_n \to L_n$ simply adds some explicit divisors and $\bar M_{0, n} \to X_n$ is a log blowup (blowup at strata), then I could compute some invariants. I may need the added divisors to be normal crossing with the preexisting divisors, which isn't true for example with the diagonal in the case $n = 5$ above.

Answer 1

Not a complete answer, too long for a comment. I think you'd find it useful to think about these things in terms of Hassett's moduli spaces of weighted pointed curves. Here's the brief version. Fix a vector of weights $\mathbf w=(w_1,\dots,w_n) \in (0,1]^n$. If $2g-2+\sum w_i > 0$ then there is a moduli space $\overline M_{g,\mathbf w}$ parametrizing $n$-pointed nodal curves satisfying a variant of the usual stability condition: markings $x_{i_1},\dots,x_{i_k}$ are allowed to coincide as long as $w_{i_1}+\dots+w_{i_k}\leq 1$. The spaces $\overline M_{g,\mathbf w}$ are smooth stacks, stratified by topological type. The space $(0,1]^n$ is decomposed into cells by the family of all hyperplanes of the form $w_{i_1}+\dots+w_{i_k}= 1$. The space $\overline M_{g,\mathbf w}$ depends only on which cell $\mathbf w$ lies in. When the parameter $\mathbf w$ crosses a hyperplane, the space $\overline M_{g,\mathbf w}$ is changed by a blow-up in a locus given by a closed stratum, and this closed stratum is isomorphic to a smaller space $\overline M_{g,\mathbf w'}$ . Namely, the blow-up locus is defined by the condition that a number of points coincide, and $\mathbf w'$ is obtained from $\mathbf w$ by replacing the corresponding entries of the weight vector with their sum. In particular:

• Each blow-up is explicitly a blow-up in a stratum for the stratification by topological type
• The blow-up loci are described purely combinatorially.

Now the point is that if $g=0$ and $\mathbf w = (a,a,\dots,a,1)$, where there are $n-1$ entries $a$ and $a=\frac{1}{n-1}+\varepsilon$, then $\overline M_{0,\mathbf w} \cong \mathbb P^{n-3}$. Kapranov's construction of $\overline M_{0,n}$ as an iterated blow-up of $\mathbb P^{n-3}$ can be understood as continuously increasing the weight parameter $\mathbf w$ to $(1,1,\dots,1)$ and keeping tracks of which blow-ups happen along the way.

Also the Losev-Manin spaces can be described this way, taking $\mathbf w = (b,b,\dots,b,1,1)$ where $b = 1/(n-2)$. This gives the factorization of $\overline M_{0,n}\to\mathbb P^{n-3}$ through the Losev-Manin space: when continuously increasing the weight parameters, you can choose to do so in a way that passes through the Losev-Manin case.

Keel's construction can also be described in these terms, using the weight vector $\mathbf w=(\varepsilon,\dots,\varepsilon,1,1,1)$, which gives $\overline M_{0,\mathbf w} \cong (\mathbb P^1)^{n-3}$.

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You also asked about to what extent the order of blow-ups matters. A useful reference for this question is Li Li: "Wonderful compactification of an arrangement of subvarieties".