I am trying to work through the proof of Bertini's Theorem from Vakil's notes. (sectin 12.4.3 to be precise).
The setup is $X=V^+(f_1,f_2,\dots ,f_r)\subset \mathbb P^n_k$ is a smooth projective variety of dimension $d$. I am looking at $Z:=\{ (p,H)\in X\times_k {\mathbb P^n_k}^* | p\in H, X\cap H \text{ is not smooth at p or }H\supseteq X \}$ where ${\mathbb P_k^n}^*$ is the dual projective space.
Then it is not very hard to see this is equivalent to looking at
$Z:=\left \{ (p,[a_0:a_1:a_2:\dots:a_n])\in X\times_k {\mathbb P^n_k} \bigg | \sum_{j=0}^n a_jp_j=0, \operatorname{corank} \begin{bmatrix} \frac{\partial f_1}{\partial X_0}(p) &\dots& \frac{\partial f_r}{\partial X_0}(p)&a_0\\ \vdots &\ddots &\vdots &\vdots\\
\frac{\partial f_1}{\partial X_n}(p)&\dots &\frac{\partial f_r}{\partial X_n}(p)& a_n \end{bmatrix} \leq \dim X \right \}$
I want to show $\dim Z\leq n-1$. At this point, Vakil fixes a point $p\in X$ and defines
$W_p= \left \{H\in {\mathbb P_n^k}^* \bigg| p\in H, X\cap H \text{ is not smooth at p or }H\supseteq X \right \}$
He says $W_p$ is defined by $d+1$ linear constraints and hence is the projective space of $\dim n-d-1$. This is where I have a problem. Since $X$ is smooth, we know $\operatorname{corank} \begin{bmatrix} \frac{\partial f_1}{\partial X_0}(p) &\dots& \frac{\partial f_r}{\partial X_0}(p)\\ \vdots &\ddots &\vdots \\
\frac{\partial f_1}{\partial X_n}(p)&\dots &\frac{\partial f_r}{\partial X_n}(p) \end{bmatrix} =\dim X$
So $\operatorname{corank} \begin{bmatrix} \frac{\partial f_1}{\partial X_0}(p) &\dots& \frac{\partial f_r}{\partial X_0}(p)&a_0\\ \vdots &\ddots &\vdots &\vdots\\
\frac{\partial f_1}{\partial X_n}(p)&\dots &\frac{\partial f_r}{\partial X_n}(p)& a_n \end{bmatrix} \leq \dim X$ gives me $d$ independent linear equations.
I presumed the other equation is given by $p_0a_0+p_1a_1+\dots +p_na_n=0$ but we have
$\sum_{j=0}^n p_j\frac{\partial f_i}{\partial X_j}(p)=\deg f_i \cdot f_i(p)=0$ so this equation turns out to be redundant. What am I missing ?
EDIT: It seems to me that $W_p = \{H\in {\mathbb P^n_k}^* \ | \ T_pX\subset H \}$ In this case, $W_p$ is the space of hyperplanes in the vector space $\frac{k^{n+1}}{T_pX}$ which is isomorphic to ${\mathbb P^{n-d}_k}^*$.
Best Answer
So I finally figured it out moments after posting the question. The trouble is my application of the Jacobian criterion. Since I am working with the affine cone, the correct condition should be $\operatorname{corank}\leq \dim X+1$ and then everything falls in place.