Let $f:X\to Y$ be a flat and proper morphism between two smooth algebraic complex varieties. The fibres $X_y$ are projective varieties of the same dimension $d$. The discriminant locus of $f$ is classically defined as:
$$\Delta(f):=\{y\in Y\colon X_y \text{ is singular} \}$$
What do we know about the basic geometric properties of $\Delta(f)$? Is $\Delta(f)$ a subvariety of $Y$? What is its dimension?
I only know that if $Y$ is a curve, then $\Delta(f)$ is a finite set of points, i.e. the support of a divisor of $Y$.