It is sometimes possible to determine the singular locus of a variety $X$ by looking only at the fibers of a resolution of singularities $\pi : Y \to X$. For instance, if the locus contracted by $\pi$ is of codimension two or greater (a small contraction), then we know that the image of $\pi$ is the singular locus of $X$.
Consider the variety $$Q := V(x_1 y_1 + x_2 y_2 + x_3 y_3) \subset \mathbb{P}^3 \times \mathbb{P}^3$$
Write $L := [1:0:0:0] \times \mathbb{P}^3$ and $L' := \mathbb{P}^3 \times [1:0:0:0]$. $Q$ is singular at $P: = ([1:0:0:0],[1:0:0:0]) = L \cap L'$.
Consider the resolutions $\pi : Y \to Q$ and $\pi' : Y' \to Q$ which are the blowings-up of $Q$ at $L$ and $L'$, respectively. (Let's agree to pretend that we do not know what the singular locus of $Q$ is, and that we've just been handed these resolutions as abstract morphisms with the knowledge that $Y$ and $Y'$ are smooth but no particular information about how they were obtained.)
Because $\pi$ is one-to-one along $L' \setminus P$, we know that $Q$ is smooth there. Similarly, we can conclude that $Q$ is smooth along $L \setminus P$. From this alone we can conclude that $Q$ is either smooth or singular at $P$ alone.
Question: Is it possible to continue this type of argument and conclude, from information about the fibers of $\pi$ and $\pi'$ alone, that $Q$ is in fact singular at $P$?