Timeline for To whom is Bézout's theorem for varieties due?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 8, 2023 at 10:38 | vote | accept | H A Helfgott | ||
| Sep 22, 2022 at 16:21 | comment | added | H A Helfgott | Right, the introduction to section I.7 feels ambiguous. Perhaps he felt that the difficult part was classical, but couldn't pinpoint whether or where the result he expected to be true and doable had been proved? | |
| Sep 22, 2022 at 12:57 | comment | added | red_trumpet | It's not clear to me if Hartshorne intended to claim a general case of Bézout's theorem was proven (which would mean that it predates Fulton's work in the early 80s), or if he was simply expecting such a theorem. He writes carefully "we should have...". About the appendix: I think Hartshorne meant a general definition of intersection multiplicity is given in the appendix (p.427), not a generalized Bézout. | |
| Sep 22, 2022 at 8:51 | comment | added | H A Helfgott | The funny thing is that, in the introduction of \S I.7 of Hartshorne, one can find (a) the general claim (as an inequality, with multiplicities) (b) the statement that "The hardest part of this generalization is the correct definition of intersection multiplicity", and that that was done by Severi (geometrically) and Chevalley and Weil, (c) a promise that, while \S I.7 treats only the case of $V_2$ a hypersurface, the general case will be treated in Appendix A... but then Appendix A seems to drop the ball, or at least I can't find generalized Bézout there. | |
| Sep 22, 2022 at 8:48 | comment | added | H A Helfgott | In the light of Fulton's remarks (see D. Dona's reply below), would the following be best? "This is a generalization of Bézout's theorem for curves, due to Fulton and Macpherson (1980) ([Fulton, Ex. 8.4.6]; see also [Vogel]. The special case where $V_2$ is a hypersurface is classical [Hartshorne, Thm. I.7.7]." | |
| Sep 22, 2022 at 8:28 | vote | accept | H A Helfgott | ||
| Sep 22, 2022 at 8:51 | |||||
| Sep 21, 2022 at 13:43 | answer | added | D. Dona | timeline score: 7 | |
| Sep 20, 2022 at 15:35 | comment | added | red_trumpet | This is example 8.4.6 in Fulton's Intersection Theory. It might be worthwhile to take a look at the Notes and References section of chapter 8, there are named many people who worked on versions of Bezout's theorem. | |
| Sep 20, 2022 at 8:21 | comment | added | H A Helfgott | A friend me tells me that Hilbert and Serre deserve credit. I'd love to give it - and I would also like to get to the bottom of the issue! | |
| Sep 19, 2022 at 14:03 | history | asked | H A Helfgott | CC BY-SA 4.0 |