Timeline for When is a map from a logarithmic tangent bundle to a normal bundle surjective?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 19, 2016 at 15:50 | history | bounty ended | Simon Wadsley | ||
| Feb 17, 2016 at 17:19 | vote | accept | Simon Wadsley | ||
| Feb 17, 2016 at 13:34 | history | edited | JoS | CC BY-SA 3.0 |
added necessary hypothesis that $X$ is smooth and elaborated on possibility to drop the smoothness of $Y$
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| Feb 17, 2016 at 13:32 | comment | added | JoS | I guess the comments about the case that $Y$ is not smooth don't hurt and I would leave them in the answer. However, as the reference above works with $X$ a complex manifold, I am unsure what happens if $X$ is not smooth. I will thus add this hypothesis to my answer. | |
| Feb 17, 2016 at 13:17 | comment | added | Simon Wadsley | Thanks again. In fact this has made me realise that my question wasn't quite stated correctly. I really wanted to assume $Y$ is smooth rather than $X$. I think this is what I need for my claim that $\iota^\ast \mathcal{T}_X\to \mathcal{N}_{Y/X}$ is surjective. I'll edit the question. If you want to make appropriate changes/delete comments about smoothness then I'll do the same. | |
| Feb 17, 2016 at 10:04 | comment | added | JoS | To see this last claim you might want to take a look at Proposition 21.2.16 of Vakil's notes math.stanford.edu/~vakil/216blog/FOAGdec2915public.pdf , where the conormal sheaf of a regular embedding is described explicitly. | |
| Feb 17, 2016 at 7:33 | comment | added | JoS | If $Y$ is not smooth, it is not clear what $T_p Y$ means (at least it is not a vector space of dimension $\dim Y$). So I guess here one must be careful how to phrase the transversality condition. However, I think that the proposal at the end of the answer should work, because then tangent vectors $v_1, \ldots, v_l$ to $X_k$ with the matrix of differentials $(\partial_{v_i} x_j)_{i,j=1, \ldots, l}$ having full rank should restrict to a basis of $\mathcal{N}_{Y/X}|_p$. | |
| Feb 16, 2016 at 20:29 | comment | added | Simon Wadsley | I see. Thanks for that reference. I'm not sure I understand where you are using $Y$ is smooth as a stronger statement than the statement that $\iota^\ast\mathcal{T}_X\to \mathcal{N}_{Y/X}$ is itself surjective. | |
| Feb 16, 2016 at 14:14 | comment | added | JoS | No, these k should be the same. My reference is Section 2 of the paper arxiv.org/abs/math/0210263 . From what is contained there, it should follow that we can choose the coordinates around $p$ such that $D$ is given by the equation $z_1, \ldots, z_k$. | |
| Feb 16, 2016 at 13:56 | comment | added | Simon Wadsley | Thanks. This looks helpful. I don't have time today to think further about it but will do so tomorrow. One quick question: are you overloading $k$ when you say that 'At a point $p\in Y\cap X_k$ the divisor $D$ has equation $z_1\ldots z_k=0$ in local coordinates'? | |
| Feb 16, 2016 at 12:46 | history | answered | JoS | CC BY-SA 3.0 |