Where are the critical points of a proper faithfully flat morphism

Question

Suppose that $X$ and $Y$ are compact complex manifolds and $f:X\to Y$ is a faithfully flat map. This map will generally not be a submersion, but it is a submersion away from singular fibres. Assuming that the singular fibres of $f$ are relatively tame (e.g. they form a normal crossing divisor in $X$), can we say anything about where critical points would be located in the singular fibres?

The naive thought is that the critical points should lie in the singular loci of the singular fibres, but does anyone know of any results that could back up that intuition?

Answer 1

Take a point $x \in X$ and set $y = f(x) \in Y$. Let $X_y = f^{-1}(y)$ the "scheme-theoretic" fiber, i.e. the complex-analytic subspace of $X$ which is cut out the equation $f(x) = y$. Your question is local in $X$, so choose complex coordinates $x_1, \dotsc, x_n$ around $x$ and $y_1, \dotsc, y_m$ around $y$. Then both assertions

1. $x$ a regular point of $f$,
2. $X_y$ is smooth at $x$,

are equivalent to

3. the differential matrix $Df(x) = \left(\frac{\partial f_i}{\partial x_j}(x)\right)_{ij}$ has full rank.

So yes, a point $x$ is critical if and only if the fiber is singular at $x$.