If $X$ is an integral, projective scheme over a ring $A$, $Z\subset X$ a nonempty, reduced, closed subscheme, and $U = X\setminus Z$, can $U$ detect whether the $Z$ is reducible? What about $Z$ smooth or nonsingular?
What about the special cases: (1) $Z$ is pure codimension 1 (2) $X = \mathbb{P}^n_k$
To elaborate, if we assume (1), then I believe $U$ will be affine, in which case we now have two rings $\Gamma(Z)$ and $\Gamma(U)$. Given $X$, each of $Z$ and $U$ (with its embedding) determines the other, so we might hope that certain information in $\Gamma(Z)$ is reflected in $\Gamma(U)$. I am interested in any kind of information that is reflected, but reducibility was the first specific property I was considering.
The presence of zero-divisors in $\Gamma(Z)$ determines its reducibility. Is there a corresponding property in $\Gamma(U)$?
I guess I am vaguely thinking of $X$ as some kind of dualizing space, which is why I figured it would need to be sufficiently nice - certainly projective, and possibly even projective space itself.