Timeline for What is $f^*TX$ for a general morphism $f\colon\mathbb{P}^1\to X$?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 9, 2020 at 10:07 | comment | added | Christoph Mark | The "example" is general enough for me and contains everything I need. | |
| Dec 9, 2020 at 10:02 | comment | added | Christoph Mark | I looked at your paper with de Jong, He. Can you please put your finger on the passage which contains the computation? Further, can you please answer my question on globally generated vector bundles on $\mathbb{P}^1$? It would be of great help... | |
| Nov 26, 2020 at 19:16 | comment | added | Jason Starr | You mention examples of morphisms that are orbit closures (you mention $\textbf{SL}_2$ orbits, but these will also be orbit closures for the action of the $1$-dimensional tori in $\textbf{SL}_2$). For these morphisms, you can compute the integers by equivariant localization. A computation of this kind is the key computation in my (unique) article with Johan de Jong and Xuhua He. I suspect that your description the pullback of the tangent bundle is correct. I will think about this and hopefully reply soon. | |
| Nov 25, 2020 at 5:08 | comment | added | Christoph Mark | I worked out an explicit description of this general morphism in my publications. In the end, I ask about one explicit morphism. Please see arxiv.org/search/… | |
| Nov 25, 2020 at 4:54 | comment | added | Christoph Mark | Globally generated in our context means probably that all the numbers in the decomposition are nonnegative. Is that right? | |
| Nov 25, 2020 at 4:51 | comment | added | Christoph Mark | Thanks for the reaction and the reference @Jason Starr. I'm asking for a general morphism, i.e. one which is in an open dense subset of the Hom scheme. I guess that it is what you mean with geometric generic point. Please take a look at the related question which popped up in the sidebar after I posted. The Grothendieck decomposition is somehow deformation invariant; it doesn't matter which morphism in the open dense set you choose, the decomposition stays the same. Sorry, if I express myself unclear, I cannot do better. | |
| Nov 24, 2020 at 17:42 | comment | added | Jason Starr | I realize now that I may have misunderstood the question. Are you asking about the pullback of the tangent bundle for every morphism from the projective line, or are you asking about the pullback only for the morphism corresponding to the geometric generic point of the Hom scheme? | |
| Nov 24, 2020 at 11:48 | comment | added | Jason Starr | Your guess is incorrect. There are many different possibilities for the pullback. One common feature is that the pullback is globally generated. For more about this, see the article of Coskun — Riedl and citations there. | |
| Nov 24, 2020 at 6:39 | history | asked | Christoph Mark | CC BY-SA 4.0 |