Timeline for Realizing algebraic curves as complete intersections
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 27, 2019 at 4:27 | comment | added | abx | @Eoin: No, the computation of the canonical bundle does not necessitate any assumption on the hypersurfaces. | |
| Nov 14, 2018 at 16:48 | history | edited | Zach Teitler | CC BY-SA 4.0 |
added 11 characters in body
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| Jul 24, 2018 at 4:30 | comment | added | abx | No, it is still valid for a singular curve $X$ if you have in mind the arithmetic genus, that is $\dim H^1(X,\mathcal{O}_X)$. For singular curves there is also the geometric genus, that is, the genus of the normalization of $X$; it is equal to the arithmetic genus minus the contribution of the singularities. | |
| Jul 23, 2018 at 14:53 | comment | added | Alm | Does the above-given equation for the genus hold only for polynomials that do not have singularity? | |
| Sep 5, 2017 at 18:40 | comment | added | roy smith | The answer of abx of course also reveals that the canonical bundle of a c.i. curve of genus ≥ 2 is very ample, hence no hyperelliptic curve of any genus ≥ 2 can occur as a c.i. This is just to illustrate explicitly the remark of abx that even among the allowable genera, exceptions exist. | |
| Feb 9, 2016 at 18:45 | comment | added | pbelmans | There is now an entry in the OEIS giving the numbers up to a certain cutoff, see oeis.org/A266322. | |
| Dec 27, 2015 at 17:22 | comment | added | abx | You are absolutely right. Sorry I didn't pay too much attention to the precise numbers -- I just wanted to indicate the principle. | |
| Dec 27, 2015 at 13:41 | comment | added | pbelmans | Shouldn't 6 also be in that list, using a planar curve of degree 5? I also think 12 should not be in the list, whilst 13 should (using (3,2,2)) and so does 15 (as a curve of degree 7). | |
| Aug 30, 2014 at 5:59 | comment | added | abx | Yes. Sorry I went too fast! | |
| Aug 30, 2014 at 4:26 | comment | added | Noam D. Elkies | You write "$g=1,\ldots,5,9,10,12,16,\ldots$", but $g=2$ does not occur (while $g=0$ of course does). | |
| Aug 29, 2014 at 20:20 | comment | added | Robin | Thanks! I feel a little foolish to have not realized (2)... | |
| Aug 29, 2014 at 20:16 | vote | accept | Robin | ||
| Aug 29, 2014 at 20:14 | history | answered | abx | CC BY-SA 3.0 |