Timeline for Classification of (complex algebraic) vector bundles on punctured affine space
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Feb 16, 2021 at 14:57 | comment | added | Jason Starr | @MichaelThaddeus For the second question, there are $S_+$-primary ideals generated by a regular sequence that are not homogeneous for any grading: $(x_0,\dots,x_{n-2},x_{n-1}-x_n^2-x_n^3,x_n^4)$ is one example. | |
| Feb 16, 2021 at 14:41 | comment | added | Jason Starr | @MichaelThaddeus Happy New Year, and I hope that you are doing well. "One thing I don't know ..." The answer to the first question is positive and follows from the "graded Nakayama's Lemma". Here is a link to a MSE answer that explains this: math.stackexchange.com/questions/557402/… | |
| Feb 7, 2021 at 22:06 | comment | added | Michael Thaddeus | One thing I don't know is whether $\pi^*E$ is trivial if and only if $E$ is a sum of line bundles, where $\pi: A^{n+1}\backslash 0 \to P^n$ is the projection. Another is whether all vector bundles on $A^{n+1}\backslash 0$ are pulled back from some weighted projective space. But I have not given these questions any thought. | |
| Feb 7, 2021 at 21:55 | comment | added | Michael Thaddeus | (4) The example given by Jason is not homogeneous for the usual grading, but it is homogeneous if $x_{n-1}$ has degree 2, so the bundle is pulled back from a weighted projective space. | |
| Feb 7, 2021 at 21:54 | comment | added | Michael Thaddeus | (3) The tangent bundle of $P^n$, pulled back to $A^{n+1} \backslash 0$, is $Z_r(f)$ in the above for $f = (x_0, \dots, x_n)$; | |
| Feb 7, 2021 at 21:54 | comment | added | Michael Thaddeus | (2) Reflexive sheaves always extend uniquely over codimension $\geq 2$ loci in smooth varieties (see Okonek-Schneider-Spindler 1.1.12); | |
| Feb 7, 2021 at 21:54 | comment | added | Michael Thaddeus | Several elementary points are worth stating explicitly: (1) Algebraic line bundles always extend uniquely over codimension $ \geq 2$ loci in smooth varieties (just take closures of the subvarieties representing the Cartier divisor class); | |
| Mar 12, 2017 at 13:30 | vote | accept | Qfwfq | ||
| Mar 10, 2017 at 16:10 | history | edited | Jason Starr | CC BY-SA 3.0 |
added 519 characters in body
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| Mar 10, 2017 at 16:05 | comment | added | Jason Starr | The rank of $M$ equals $n$. Since the construction above only works for $n\geq 2$ (so that we know that $Z_{n-1}(\underline{f})$ is reflexive), it does not give examples of line bundles. | |
| Mar 10, 2017 at 16:03 | comment | added | Qfwfq | Interesting. What about line bundles? Can $M$ ever have rank $1$? | |
| S Mar 10, 2017 at 15:46 | history | answered | Jason Starr | CC BY-SA 3.0 | |
| S Mar 10, 2017 at 15:46 | history | made wiki | Post Made Community Wiki by Jason Starr |