Exercise in Beauville, about the blowup of $P^2$ at seven points I am trying to solve: (this is an exercise in Beauville's book, Complex Algebraic Surfaces, page 52.)
Consider $7$ points of $P^2$ in general position. Let $P_7$ denote the blowup of $P^2$ at these points and $H$ the strict transform of a cubic containing them. Show that $|H|$ defines a two-to-one map $P_7 \to P^2$ branched along a smooth quartic.
I showed that $|H|$ is base-point-free, so there is a well-defined map $P_7\to P^2$, but I can't show the rest. How can we show that this map is branched along a quartic?
 A: Expanding on my comment because I figured out what errors I was making before: let $L$ denote the pullback of the class of a general line in $P^2$ to $P_7$. The degree of a finite map over $\mathbb C$ is the cardinality of a general fiber. Since a general point is the intersection of two general lines, and H is by definition the class of the pullback of $\mathcal O_{P^2}(1)$, we can compute this as the intersection number
$$
H^2 = (3L - E_1 - \cdots -E_7)^2 = 9L^2 + E_1^2 + \cdots + E_1^2 = 9-1-\cdots -1=2,
$$
where $L.E_i$ and all cross terms $E_i.E_j$ vanish because $L$ is general, and because blowing up general points produces disjoint exceptional divisors.
As for the branching, this is essentially a topological question. The topological Euler characteristic $\chi$ of $P^2$ is $3$, and blowing up a point (since it replaces a point with a $2$-sphere) increases $\chi$ by $1$, so $\chi(P_7) = 10$. Now when we have a branched double cover $X\to Y$, we can think of $X$ as being constructed by cutting away the branch locus $C$ from $Y$ to get an affine variety $U$, taking two copies of $U$, and gluing back together along a single copy of the branch locus, so $\chi(Y) = 2\chi(U) + \chi(C)$. Filling in what we know we get
$$
10 = 2(3 - (2-2g(C))) + (2-2g(C)) = 2(1 + 2g(C)) + 2 - 2g = 4 + 2g(C)
$$
hence $g(C) = 3$. Now assuming that $C \subset P^2$ is smooth, it must be a quartic by the degree-genus formula.
To see that $C$ is smooth requires a local calculation that I will leave to you. You want to show that the surface defined by $w^2 = F(x,y,z)$ in $P^2 \times A^1$ is smooth if and only if the plane curve $F=0$ is smooth. Since we know $P_7$ is smooth, we conclude that $C$ is smooth.
