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Let $X\subset \mathbb{P}^5$ be a quintic del Pezzo surface embedded anti-canonically and suppose $X$ is smooth. Suppose further we are given a line $L\subset X$. After a suitable change of variables we may assume that $L$ is given by $x_2=\cdots = x_5=0$ if we work with coordinates $x_0,\dots, x_5$ on $\mathbb{P}^5$. Pick a point $p\in L$ and consider the projection $\varphi\colon X\setminus p \to \mathbb{P}^4$. For example, if $p=(1\colon 0 \colon 0\colon 0\colon 0\colon 0)$ this map is given by $(x_0\colon x_1\colon \cdots \colon x_5)\mapsto (x_1\colon x_2\colon \cdots \colon x_5)$. The closure of the image of $\varphi$ is a quartic surface $Y$ since $p$ is a non-singular point of $X$.

My suspicion is that $Y$ is a del Pezzo surface of degree 4 with a singularity at $(1\colon 0 \colon \cdots \colon 0)$. By this I mean that the anti-canonical divisor of $Y$ is very ample and has self intersection number 4.

It follows for example from Corollary 24.5.2 in Manin's "Cubic surfaces" that $Y$ is again a del Pezzo surface. The projection looks to me related to blowing-up $X$ in $p$, which would yield a birational surface whose self intersection number is 4, but I am not sure about a precise relation.

Does anyone know a reference where this situation is studied? Any help would be greatly appreciated!

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The map, indeed, blows up the point $p$ and then contract the strict transforms of all lines passing through $p$ (which are $(-2)$-curves). Therefore, $Y$ is a singular del Pezzo surface of degree $4$ with one or two nodes, depending on whether the point $p$ lies on one line of $X$ or on two lines.

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  • $\begingroup$ Thanks for the quick answer! Do you have a reference where this is studied? $\endgroup$
    – MightyGuy
    Jun 19, 2023 at 12:50
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    $\begingroup$ This should be very standard, but I don't know a reference. $\endgroup$
    – Sasha
    Jun 19, 2023 at 13:38
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    $\begingroup$ @Sasha Familiar words... $\endgroup$
    – R.P.
    Jun 19, 2023 at 13:53

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