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For reference, see the image below for the exercise. I have been able to solve (a) and (b), but (c) is giving me trouble.

First off, let me assume that $P=(0,0,1)$ so that the linear system $\mathfrak{d}$ is spanned by $x^2,y^2,xy,xz,yz$. It is not hard to see that $\mathfrak{d}$ gives an immersion $X-P\rightarrow\mathbb{P}^4$.

For the rest, I cannot seem to provide a convincing proof. Hartshorne studies how one can obtain a closed embedding and prove that the surface we get is of degree $3$ in Section V.4. but I cannot follow the chapter too well.

So far, I only have heuristics on how to do the last part. I know that $\mathbb{P}^2$ can be covered by lines going through $P$ (which I think I have a proof for). Then if "blowing-up separates lines through points" then I can imagine that the lines are sent to nonintersecting lines. Since the $\pi:\widetilde{X}\rightarrow X$ is a surjective morphism by Proposition II.7.16, I think this would imply that $\widetilde{X}$ is a union of these lines.

My question(s): Can someone provide a hint on how to formalize my idea in the last paragraph? Furthermore, how does the map from $U$ extend to a map from $\widetilde{X}$?

7.7. Some Rational Surfaces. Let $X=\Bbb P^2_k$, and let $|D|$ be the complete linear system of all divisors of degree 2 on $X$ (conics). $D$ corresponds to the invertible sheaf $\mathcal{O}(2)$, whose space of global sections has a basis $x^2,y^2,z^2,xy,xz,yz$, where $x,y,z$ are the homogeneous coordinates of $X$.

  • (a) The complete linear system $|D|$ gives an embedding of $\Bbb P^2$ in $\Bbb P^5$, whose image is the Veronese surface (I, Ex. 2.13).
  • (b) Show that the subsystem defined by $x^2,y^2,z^2,y(x-z),(x-y)z$ gives a closed immersion of $X$ into $\Bbb P^4$. The image is called the Veronese surface in $\Bbb P^4$. Cf. (IV, Ex. 3.11).
  • (c) Let $\mathfrak{d}\subset|D|$ be the linear system of all conics passing through a fixed point $P$. Then $\mathfrak{d}$ gives an immersion of $U=X-P$ into $\Bbb P^4$. Furthermore, if we blow up $P$, to get a surface $\widetilde{X}$, then this map extends to give a closed immersion of $\widetilde{X}$ in $\Bbb P^4$. Show that $\widetilde{X}$ is a surface of degree 3 in $\Bbb P^4$, and that the lines in $X$ through $P$ are transformed into straight lines in $\widetilde{X}$ which do not meet. $\widetilde{X}$ is the union of all these lines, so we say $\widetilde{X}$ is a ruled surface (V, 2.19.1).
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Hint for showing that $\widetilde{X}\to\Bbb P^4$ is a closed immersion: show that there's a linear inclusion from $\Bbb P^4\to\Bbb P^5$ so that the composite is a closed immersion you've seen before, and conclude that $\widetilde{X}\to\Bbb P^4$ must have been a closed immersion via exercise II.4.8 (more details in a spoiler below).

Further hint:

$\widetilde{X}$ embeds as a closed subscheme of $\Bbb P^2\times\Bbb P^1$, which embeds as a closed subscheme of $\Bbb P^5$ by the Segre embedding. Try to factor $\widetilde{X}\to\Bbb P^2\times\Bbb P^1\to\Bbb P^5$ in to $\widetilde{X}\to\Bbb P^4\to\Bbb P^5$ where the last map is a linear inclusion. You will need some explicit work with coordinates here, but it should be relatively straightforward.


For determining the degree and showing that distinct lines through $P$ get turned in to straight lines that don't meet, the key is to understand what happens geometrically in the blowup. The blowup separates tangent vectors above the blown-up point, so for two smooth curves meeting transversely at $P$, their strict transforms do not meet in $\widetilde{X}$. To show this, you can work affine-locally: suppose the local equation of your lines are $ax+by$ and $cx+dy$. Then (assuming that $b,d\neq0$, which can always be done by a coordinate change) in the blowup chart obtained by setting $y=tx$, the strict transforms of these lines are given by $a+bt$ and $c+dt$, which do not intersect if the original lines did not. A similar argument for conics will let you compute the degree of $\widetilde{X}\subset\Bbb P^4$ by viewing the intersection of hyerplanes in $\Bbb P^4$ with $\widetilde{X}$ as conics on $X$. Choosing them appropriately, you can use the same computation along with theorem I.7.7 and some knowledge about how quadrics intersect in $\Bbb P^2$ to get the degree.

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  • $\begingroup$ Thanks for the hint. I'll have to think about it. In terms of showing that $\widetilde{X}$ is a union of the lines we pull-back, is there a way to see that from your answer? Does my argument just hold verbatim once I establish that the strict transforms of lines through $P$ no longer intersect? $\endgroup$
    – Shrugs
    Feb 27, 2022 at 7:07
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    $\begingroup$ Basically? First, every point on $X$ is on a line through $P$: for points in $U$, take the line through the point and $P$, and for a point on the exceptional divisor, take the line through $P$ with that tangent direction. Next, the strict transforms of distinct lines through $P$ do not intersect by the argument in the post. So (on the level of closed points) $\widetilde{X}$ is the union of the strict transforms of lines through $P$ and no two distinct members of that set meet eachother. That last sentence is not a rigorous claim for you to prove, it's a "this will be useful later". And it is! $\endgroup$
    – KReiser
    Feb 27, 2022 at 9:14
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    $\begingroup$ Essentially everything you see in II.7 is basic setup for doing concrete algebraic geometry with curves and surfaces in chapters IV and V. It's great and necessary if you're going to get in to those parts of the book, but it can maybe feel a little "ok so what" if you don't tie it to that material or other examples. $\endgroup$
    – KReiser
    Feb 27, 2022 at 9:16

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