Timeline for Normal bundle to Veronese varieties $v_d(\mathbb{P}^n)$ into $\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^n}(d)))$
Current License: CC BY-SA 4.0
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| Jul 16, 2021 at 11:52 | comment | added | Sasha | @gigi: Yes, indeed, the first realization is a combination of the two Euler sequences. The second part can be deduced as follows. The Euler sequence gives a filtration on $V \otimes \mathcal{O}(1)$ with factors $\mathcal{O}$ and $T$. Taking its symmetric power one gets a filtration on $S^dV \mathcal{O}(d)$ with factors $S^iT$, $0 \le i \le d$. Taking the quotient one gets the second part of the answer. | |
| Jul 16, 2021 at 11:47 | comment | added | Sasha | @JasonStarr: Jason, thanks for the correction! | |
| Jul 16, 2021 at 11:46 | history | edited | Sasha | CC BY-SA 4.0 |
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| Jul 16, 2021 at 11:07 | comment | added | Jason Starr | Typo correction: in the displayed equation for the associated graded pieces, the summands should be the $i$-fold symmetric powers of the tangent bundle, not the $d$-fold symmetric powers. Also, one frequently useful observation that is not so clear from this answer is that the normal bundle is a subbundle of a direct sum of copies of $\mathcal{O}(2d)$. This comes from the standard quadratic generators for the homogeneous ideal of the Veronese variety. | |
| Jul 16, 2021 at 9:03 | comment | added | gigi | thank you, now it's clear. I guess that the first realization of $N$ has to do with the standard normal sequence and Euler sequence. Could you give me a reference where I can find the construction of $N$ you suggested in the second part of the answer? | |
| Jul 16, 2021 at 9:01 | vote | accept | gigi | ||
| Jul 16, 2021 at 4:38 | history | answered | Sasha | CC BY-SA 4.0 |