What is the Theorem on Coherent Cohomology and Base Change good for?

One version of the theorem is:

Suppse $f \colon X \to Y$ is a proper morphism of noetherian schemes and $F$ is a $Y$-flat coherent sheaf on $Y$. If $y_0 \in X$ and $q \in \mathbb{Z}$ have the property that the natural map

$$\alpha^{q}(y_0) \colon k(y_0) \otimes R^{q}f_{*}(F) \to H^{q}(X_{y_0}, F/\mathfrak{m}_{y_0}F)$$

is surjective, then there exists a neighborhood $U \subset Y$ with the property that $\alpha^{q}(y)$ is an isomorphism for all $y \in Y$. Furthermore, $\alpha^{q-1}(y_0)$ is surjective if and only if $R^{q}f_{*}(F)$ is locally free of finite type in a neighbohood of $y_0$.

**What are some results that make use of this theorem?**

Here are the applications I have seen:

1) Top-dimensional cohomology commutes with base change:

If the fibers $X_{y}$ all have dimension at most $n$, then trivially $\alpha^{n+1}(y_0)$ is surjective and $R^{n+1}f_{*}(F)$ is locally free of finite type. Thus, $\alpha^{n}(y_0)$ is always an isomorphism. For example, if $f: X \to Y$ is a family of curves, then the fibers of $R^{1}f_{*} (\mathcal{O}_{X})$ are the 1st cohomology groups of the fibers of $f$.

This fact is used to construction the Hodge bundle on the Deligne--Mumford moduli stack of stable curves.

2) The direct image of the structure sheaf of a family of varieties is the sheaf of regular functions on the base:

Trivially $\alpha^{-1}(y_0)$ is surjective. Thus, if $\alpha^{0}(y_0)$ is surjective, then $f_{*}(F)$ is locally free of finite type on a neighborhood of $y_0$. When $f$ is surjective with geometrically reduced and connected fibers and $F=\mathcal{O}_{X}$, then it is not hard to show that $\alpha^{0}(y_0)$ is surjective (every global function on $X_y$ is constant). In particular, the direct image of the structure sheaf is locally free of finite type, and one can show further that it is $\mathcal{O}_{Y}$, the *trivial* locally free module of rank $1$.

When the condition

$$\mathcal{O} = f_{*}(\mathcal{O}_{X}) $$

is satisfied, it is a theorem of Artin that the relative Picard space (parameterizing line bundles on the fibers of $X_{y}$) exists.

3) The Picard bundle is a vector bundle.

Suppose $C$ is a smooth, projective curve of genus $g$ (over a field) and $Y = J^{d}_{C}$ is its degree $d$ Jacobian (parameterizing line bundles on $C$ of degree $d$). If $d \ge 2g-1$, then the Picard bundle $E$ is a vector bundle with the property the fiber over $y_0 \in Y$ is $H^{0}(C, L)$, where $L$ is the line bundle corresponding to $y_0$. If $F$ is a universal family of line bundles on $X = C \times J^{d}_{C}$, then the Theorem on Base Change implies that $E$ can be defined to be the direct image of $F$ under the projection map $f \colon X \to Y$. Indeed, both $\alpha^{-1}(y_0)$ and $\alpha^{1}(y_0)$ are always surjective as they map to the zero module.

The projectivization of the Picard bundle is the $d$-th symmetric power of the curve, and this description provides one approach to studying the symmetric power.

What are some other applications?