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Are there known explicit parametrizations of smooth del Pezzo surfaces, say of degree $5$ or higher, as surfaces cut out by equations in some projective space?

The closest I've been able to find is on p. 42 of "Quantitative arithmetic of projective varieties" by Tim Browning (equation 2.21), where he describes a singular del Pezzo surface of degree $6$ by a system of $9$ quadrics in $\mathbb{P}^6$.

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    $\begingroup$ I'm pretty sure a smooth degree $6$ del Pezzo can be described with coordinates $ b ,a_1,a_2,a_3,a_4,a_5,a_6$ and equations $b a_i = a_{i-1}a_{i+1}$ for $i$ from $1$ to $6$ and $a_i a_{i+3} = b^2$ for $i$ from $1$ to $3$. $\endgroup$
    – Will Sawin
    Commented Nov 10, 2021 at 3:26
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    $\begingroup$ Equations for the degree 5 case are in arxiv.org/abs/1812.10715. Equations in a product of lines are in arxiv.org/abs/1803.02984 section 3 $\endgroup$ Commented Nov 10, 2021 at 13:23

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A smooth del Pezzo surface $X_d$ of degree $d\geq3$ has very ample anticanonical divisor $-K_X$. It is also the blowup $X\to \mathbb P^2$ in $k=9-d$ points $P_1,\ldots,P_k\in\mathbb{P}^2_{x,y,z}$ in general position, so we have that $-K_X=\pi^*\mathcal{O}_{\mathbb P^2}(3) - E_1 - \cdots - E_k$. 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 $P_1,\ldots,P_k$. For example if $d=6$ then, up to a linear automorphism of $\mathbb P^2$, we can assume that the three points we blow up are $(1:0:0)$, $(0:1:0)$ and $(0:0:1)$ and the seven-dimensional linear system of cubic polynomials vanishing at these points has a basis given by $xyz,x^2y,xy^2,y^2z,yz^2,xz^2,x^2z$, which are the coordinates $b,a_1,\ldots,a_6$ in Will Sawin's comment above. (This also follows from the description of $X_6$ as a toric variety.)

For each $d$ this will give you an embedding $X_d\hookrightarrow \mathbb{P}^d$ of codimension $d-2$ where the $X_d$ is cut out by quadratic polynomials. (For $X_9=\mathbb P^2$ it gives the third Veronese embedding.) However there are many simpler embeddings that one can work with if you want to work with explicit equations. For example $X_6$ can also be described as a smooth hypersurface of bidegree $(1,1,1)$ in $\mathbb P^1\times \mathbb P^1\times \mathbb P^1$.

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    $\begingroup$ Possibly the OP wants to work over a non-algebraically closed field (the degree $6$ del Pezzo has a large automorphism group). $\endgroup$ Commented Nov 10, 2021 at 10:44
  • $\begingroup$ @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/… $\endgroup$ Commented Nov 10, 2021 at 18:07
  • $\begingroup$ "Therefore you can write down the anticanonical embedding of 𝑋 by writing down a basis for the linear system of cubic polynomials in π‘₯,𝑦,𝑧 that vanish at 𝑃1,…,π‘ƒπ‘˜." Can you explain this in more detail? $\endgroup$ Commented Aug 18, 2023 at 18:19
  • $\begingroup$ In which direction do you want more explanation? Why it is true? Or how to do it? $\endgroup$
    – Tom Ducat
    Commented Aug 19, 2023 at 17:41
  • $\begingroup$ 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. $\endgroup$ Commented Aug 21, 2023 at 9:04

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