Timeline for Degree of the preimage of a variety
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2023 at 11:03 | comment | added | H A Helfgott | Confusingly, generalized Bézout is stated only for hypersurfaces on the Wikipedia page on Bézout's theorem. There's a lot going on under the hood in Fulton's book! | |
| Sep 12, 2023 at 11:00 | comment | added | H A Helfgott | Update: I was just talking to an algebraic geometer last week - it all checks out. He said a more grown-up proof might go through the Chow variety. | |
| Aug 13, 2023 at 6:01 | history | edited | H A Helfgott | CC BY-SA 4.0 |
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| Aug 13, 2023 at 5:03 | history | edited | H A Helfgott | CC BY-SA 4.0 |
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| Aug 13, 2023 at 4:24 | history | edited | H A Helfgott | CC BY-SA 4.0 |
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| Aug 10, 2023 at 14:44 | comment | added | H A Helfgott | ... and that so are the statement in my answer and the proof in my answer (but everybody is obviously welcome to still tell me if there's something unclear or there is a mistake). I take a partial answer to the Sumerian meta-question is that, before Fulton and Macpherson came up with intersection theory (which is what is really under the hood of what I've called "Bézout" in the above - really generalized Bézout), giving a general result like this, valid for all varieties, would have been hard. | |
| Aug 10, 2023 at 10:17 | vote | accept | H A Helfgott | ||
| Aug 10, 2023 at 5:35 | comment | added | H A Helfgott | I don't want to turn this into the parrot sketch, but I'd say Lemma 3 is completely correct (and so is its proof). | |
| Aug 8, 2023 at 20:27 | history | edited | H A Helfgott | CC BY-SA 4.0 |
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| Aug 8, 2023 at 19:58 | comment | added | H A Helfgott | (I suspect you are tacitly using a Segre embedding, which of course is not of degree 1. As I said, let us work entirely in affine space, at least in the proof of Lemma 3.) | |
| Aug 8, 2023 at 19:50 | comment | added | H A Helfgott | Just to be clear: everything in the proof of Lemma 3 is to be understood in affine space. What is the questionable step then? | |
| Aug 8, 2023 at 19:45 | comment | added | H A Helfgott | @Libli Obviously what I meant by my remark is that this feels like very classical.stuff and I would think it has to be in the literature (assuming it is correct, obviously); I can't see how anybody could have honestly interpreted my remark as you claim to have. But let us focus on the math. | |
| Aug 8, 2023 at 19:43 | comment | added | H A Helfgott | @Libli I'm afraid I can't quite understand your example. Are you claiming you have two linear varities in $\mathbb{P}^3$ that intersect in two points? If that's not what you claim, I don't see how it affects what I've said; all I've used in that step is Bézout's theorem in its general form (as in Fulton) - and in affine space at that. | |
| Aug 8, 2023 at 14:09 | comment | added | Libli | So lemma 3 is not proven and hence your "pseudo-proof" of your claimed result is not correct. The reason this proof does not appear in any Sumerians textbooks is because it's just plain wrong, and not at even at the level of the Sumerians. Perhaps you could be slightly more modest and not dare to put yourself at the same level as Chevalley and Fulton? | |
| Aug 8, 2023 at 13:58 | comment | added | Libli | Your "proof" of Lemma3 seems to be wonky-wonky. Assume $g(V)$ is a line in $\mathbb{P}^3$, $W= \mathbb{P}^1$ and $m=1$. Then the intersection of $W \times \mathbb{A}^1$ with $g(V)$ consists of $2$ points, and so has degree $2$. While $\deg(W) \deg(g(V)) =1$. | |
| Aug 7, 2023 at 6:15 | vote | accept | H A Helfgott | ||
| Aug 8, 2023 at 20:47 | |||||
| Aug 2, 2023 at 8:16 | comment | added | H A Helfgott | @libli It has been a couple of days, so I hope the above allays your concerns. | |
| Aug 1, 2023 at 9:04 | comment | added | H A Helfgott | I'm also wondering how come this all (my self-answer included) isn't at least as old as Fulton or Chevalley or the Sumerians or what have you. | |
| Aug 1, 2023 at 8:58 | comment | added | H A Helfgott | @D.Dona I'm very happy to credit it to you, and at any rate it's all going to come out on work by both of us! | |
| Aug 1, 2023 at 8:40 | comment | added | D. Dona | I think Lemma 3 came out implicitly from my side, given the amount of time I was thinking about both Lemma 2 and this projection trick, but I'm not advancing a claim. Lemma 2 came from this much older question. | |
| Jul 31, 2023 at 8:51 | history | edited | H A Helfgott | CC BY-SA 4.0 |
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| Jul 31, 2023 at 7:56 | comment | added | H A Helfgott | Here is a proof of Lemma 3 (namely, $\deg(f^{-1}(W)) \leq \deg(V) \deg(W) D^{\dim V}$). Let $g:V\to V\times \mathbb{A}^n$ be given by $g(x) = (x,f(x))$ and let $\pi_1$ be the projection $V\times \mathbb{A}^n \to V$. Then $$f^{-1}(W) = \pi_1(g(V)\cap (\mathbb{A}^m\times W)),$$ and so, by Bézout and Lemma 2, $$\begin{aligned}\deg(f^{-1}(W)) \leq \deg(g(V) \cap (\mathbb{A}^m\times W))\leq \deg(g(V)) \deg(W) \leq \deg(V) \deg(W) D^{\dim(V)}. \end{aligned}$$ (Whether this was my own past self, my coauthor Daniele Dona or someone on MathOverflow, I cannot recall.) | |
| Jul 31, 2023 at 7:51 | history | answered | H A Helfgott | CC BY-SA 4.0 |