Hartshorne Exercise II.7.7(c) Studying a particular blow-up and ruled surface.

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).

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).
$$\widetilde{X}$$ 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.
• 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? Feb 27, 2022 at 7:07
• 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! Feb 27, 2022 at 9:14