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:

- 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 $n-1$.
- 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.
- ? 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:

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.

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?

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

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.

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