Let $k$ be an infinite field. Let $X\subseteq\mathbb P_k^n$ be a projective subvariety. Let $D$ be a Cartier divisor on $X$. Mumford and Oda’s Algebraic Geometry II says (in the paragraph following Definition 7.7.3 (on page 258 of its draft)) that
Now take a homogeneous polynomial $F$ endowed with the following properties:
a) $X\not\subseteq V(F)$ and the effective Cartier divisor $H=\operatorname{Tr}_X(V(F))$ is reduced and irreducible,
b) no component $D_j$ of $\operatorname{Supp} D$ is contained in $V (F )$.
It can be shown that such an $F$ exists (in fact, in the affine space of all $F$’s, any $F$ outside a proper union of subvarieties will have these properties).
I can’t understand why such an $F$ exists.
I think “the affine space of all $F$’s” is the affine open $X_F$ of $X$ defined by $F$. Then the property (b) for example says that $X_F\cap D_j\ne\emptyset$ for all $j$, but how can we do that? I can’t get what “an $F$ outside a proper union of subvarieties” means.
Any help would be appreciated. Thanks in advance.