Timeline for Explicit defining equations for del Pezzo surfaces
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 22, 2023 at 19:54 | comment | added | Tom Ducat | The reason it is an embedding is that $-K_X$ is a very ample divisor class for $X$. (Not true if $k>6$.) The reason you can express it in terms of cubics like that follows from the standard formula for pulling back divisors after blowing up a smooth point on a surface. | |
| Aug 21, 2023 at 9:04 | comment | added | woolly-minded | What I don't understand is why this is true: why knowing that $-K_X=\pi^*\mathcal O_{\mathbb P^2}(3)-E_1-\dots-E_k$ (which I understand) gives us the fact that we can write down the anticanonical embedding in that way. | |
| Aug 19, 2023 at 17:41 | comment | added | Tom Ducat | In which direction do you want more explanation? Why it is true? Or how to do it? | |
| Aug 18, 2023 at 18:19 | comment | added | woolly-minded | "Therefore you can write down the anticanonical embedding of 𝑋 by writing down a basis for the linear system of cubic polynomials in 𝑥,𝑦,𝑧 that vanish at 𝑃1,…,𝑃𝑘." Can you explain this in more detail? | |
| Nov 10, 2021 at 18:07 | comment | added | Nicolas Banks | @JasonStarr I am indeed interested in how the construction would generalize to, say, $\mathbb{Q}$ instead of $\mathbb{C}$. In particular, I want to take the points $P_1,...,P_k$ to be Galois conjugates over some number field of degree $k$. But this answer is a great starting point, especially the comments about the anticanonical embedding. Also to add, a discussion on the aforementioned automorphism group can be found here: mathoverflow.net/questions/212015/… | |
| Nov 10, 2021 at 10:44 | comment | added | Jason Starr | Possibly the OP wants to work over a non-algebraically closed field (the degree $6$ del Pezzo has a large automorphism group). | |
| Nov 10, 2021 at 10:00 | history | answered | Tom Ducat | CC BY-SA 4.0 |