Let $f:X\to B$ be a projective family of hypersurface of type $(d,n)$ in on some curve $B$, i.e., $X\subset B\times \mathbb P^N$ is closed subscheme, and $f:X\to B$ is flat, with every fiber isomorphic to $n$-dimensional hypersurface of degree $d$. I want to know the following:
Is it true that every fiber of $f$ admit an embedding to $\mathbb P^{n+1}$? i.e. there exists some other flat family $f':X'\to B$ with $X'\subset B\times \mathbb P^{n+1}$ a divisor, such that the two family $f$ and $f'$ are isomorphic?
I tend to believe this is not true, but I want to know a counter-example. Any comments would be helpful.
No, this is not possible. For example, let $f \colon X \to B$ be a nontrivial etale double cover. Every fiber is a union of two points, so it can be embedded into $\mathbb{P}^1$ as a hypersurface. But there is no embedding $X \subset B \times \mathbb{P}^1$.
Indeed, assume $X \subset B \times \mathbb{P}^1$. Then $X$ is a divisor of relative degree 2 over $B$. Since $Pic(B \times \mathbb{P}^1) \cong Pic(B) \oplus Pic(\mathbb{P}^1)$, the corresponding line bundle can be written as $L \boxtimes O(2)$ for some $L \in Pic(B)$, so that $X$ defines a nonzero element in $$ H^0(B \times \mathbb{P}^1, L \boxtimes O(2)) = H^0(B,L) \otimes H^0(\mathbb{P}^1,O(2)). $$ In particular, $H^0(B,L) \ne 0$.
On the other hand, we have a resolution $$ 0 \to L^{-1} \boxtimes O(-2) \to O_{B \times \mathbb{P}^1} \to O_X \to 0, $$ which implies that $$ f_*O_X \cong O_B \oplus L^{-1}. $$ Since $f$ is etale and nontrivial, the line bundle $L^{-1}$ on $B$ is a nontrivial element of order 2 in $Pic(B)$. In particular, $L \cong L^{-1}$ has no global sections.