Let $f$ be a homogenous polynomial in $z_0, ..., z_n$ of degree $d$ and consider the divisor $D$ with defining data $(U_i, f(z_0/z_i, ..., z_n/z_i))$ where $U_i$ is the affine space $\{z_i \neq 0\}$. Under what circumstances is $D$ a single irreducible hypersurface?
For example, if $f=z_0^d$, then $D=dH_0$ where $H_0$ is the irreducible hypersurface $z_0=0$, so this fails. More generally, if $f$ is not irreducible as a polynomial in $\mathbb{C}[z_0,...,z_n]$ then $D$ will also not be a single irreducible hypersurface. Does the converse hold?
My definitions for the defining data of a divisor (on a compact space $X$) are as follows: First, for an irreducible hypersurface, we have local defining functions $(U_{\alpha}, f_{\alpha})$ on a finite open cover $U_{\alpha}$ of $X$, such that $f_{\alpha}=0$ cuts out $Y$ on $U_{\alpha}$ and $f_{\alpha}$ divides $g$ in $O_{X,p}$ for all $p \in U_{\alpha} \cap Y$ and all $g$ vanishing in a neighbourhood of $p$ (I think this is equivalent to $f_{\alpha}$ being square-free in each stalk $O_{X,p}$).
Then, if $D=\sum a_i Y_i$, we pick an open cover $U_{\alpha}$ such that $Y_i$ is defined by $(U_{\alpha}, f_{\alpha,i})$, then set $f_{\alpha} = f_{\alpha, 1}^{a_1}...f_{\alpha, n}^{a_m}$.