In Hartshorne's proof of Bertini's theorem, given a linear system $|H|$, he defines the locus of "bad" hyperplanes $B_x$ for each point $x\in X\subset \Bbb P^n$ a projective variety, shows that this is a proper linear subset of $\{x\}\times |H|$, defines $B$ to be the union of all pairs $(x,H)$ so that $H\in B_x$, and then claims that "clearly $B$ is the set of closed points of a closed subset of $X\times|H|$".
I don't understand this. Clearly the statement "if a subset has proper closed intersection with all fibers of a map over closed points, then it's a proper closed subset" is false – consider the inclusion of a copy of $(\Bbb P^1\setminus\{0\})\times\{p\} \to \Bbb P^1\times\Bbb P^1$ followed by a projection. How can I rigorously see Hartshorne's claim?
Best Answer
I pieced this together after the discussion with Mindlack in the comments. Define $B$ to be the subset of $X\times\Bbb P^n$ cut out by the minors of size $n-\dim X+1$ of the matrix $$\begin{pmatrix} \frac{\partial f_1}{\partial x_0} & \cdots & \frac{\partial f_1}{x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_0} & \cdots & \frac{\partial f_m}{x_n} \\ a_0 & \cdots & a_n \end{pmatrix}$$
where $f_i$ are a generating set for $X$ and $a_j$ are coefficients for the hyperplane $V(\sum a_jx_j)$ where the $x_j$ are coordinates on $\Bbb P^n$. It's not so hard to see that the vanishing of all of these minors at a point $x\in X$ means that the dehomogenization of $\sum a_jx_j$ is in the square of the maximal ideal there via the Jacobian criterion. This exactly gives $B_x$ as the fiber of $B$ over $x$.