Timeline for Blowing-up projective spaces of parametrized rational curves
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 9, 2018 at 16:59 | vote | accept | CommunityBot | ||
| Feb 4, 2018 at 19:31 | answer | added | Jason Starr | timeline score: 3 | |
| Feb 4, 2018 at 17:22 | comment | added | Jason Starr | You are completely correct that the inverse image of $Z_d$ in $M$ has irreducible components of codimension $>1$. Thus, the morphism from $M$ to $\mathbb{P}^N$ does not factor through $X_d$, much less $X_1$. | |
| Feb 4, 2018 at 16:54 | comment | added | Jason Starr | No, I am not thinking about the wonderful compactification of $\text{SL}_{n+1}$. There is a method for "making conical compactifications wonderful" that applies, in particular, to the compactifications of semisimple groups of adjoint type studied by DeConcini and Procesi. However, the theory of MacPherson and Procesi applies to many other conical stratifications than compactifications of semisimple groups. | |
| Feb 4, 2018 at 16:48 | comment | added | user117617 | Are you thinking about the wonderful compactification of $SL_{n+1}$? If so it seems to me that this sequence of blow-ups is a different compactification. Furthermore, the morphism from $M$ to $\mathbb{P}^N$ contracts the boundary divisors to subvarities of smaller dimension than the $Z_i$. So I do not think that the morphism $M\rightarrow \mathbb{P}^N$ factors through $X_1$. | |
| Feb 4, 2018 at 16:21 | comment | added | Jason Starr | Your stratification is a "conical stratification" in the language of MacPherson and Procesi, "Making conical compactifications into wonderful ones". The iterative blowing up that you describe is what MacPherson-Procesi call the "minimal wonderful blowup". The blowing up you describe does not admit a $1$-morphism to the stack $M$ of genus $0$ stable maps to $\mathbb{P}^1\times \mathbb{P}^n$ of bidegree $(1,d)$. There is a $1$-morphism from $M$ to $\mathbb{P}^N$, and perhaps this factors through the blowing up. | |
| Feb 4, 2018 at 15:13 | history | asked | user117617 | CC BY-SA 3.0 |