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).
Best Answer
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:
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.