Here are four ``applications'' of the Grothendieck-Riemann-Roch theorem that I know of.

**1. Moduli space of Enriques surfaces**

The coarse moduli space of Enriques surfaces is known to be quasi-affine. A proof of this was given by Pappas using the Grothendieck-Riemann-Roch theorem in:

http://arxiv.org/abs/math/0701546

More precisely, it's the following result which is shown in the above article using GRR.

**Theorem.** The line bundle $R^0 f_\ast (\mathcal{L}\otimes \mathcal{L})$ is a torsion line bundle on $Y$.

**2. Computing with the multiplication map on an abelian variety**

Let $X$ be an abelian variety of dimension $g$. The following is based on the article

*Heights for line bundles on arithmetic varieties*

by J. Jahnel. (You can find it easily with Google.)

Let $p:X\times X \longrightarrow X$ be the projection onto the first coordinate. Similarly, let $q:X\times X\longrightarrow X$ be the projection onto the second coordinate. For any line bundle $\mathcal{F}$ on $X$, we define its *Mumford line bundle* on $X\times X$, denoted by $\Lambda$, as $$\Lambda := m^\ast \mathcal{F}\otimes (p^\ast \mathcal{F})^{-1} \otimes (q^\ast \mathcal{F})^{-1}.$$ The following theorem is a special case of Theorem 1.7 in Jahnel. Its proof uses GRR and is contained in the proof of Proposition 3.4.

**Theorem.**
For any ample line bundle $\mathcal{L}$, we have that $$(\det q_!(\Lambda\otimes p^\ast\mathcal{L}))^{-1} = \left(\det q_!(m^\ast\mathcal{L} \otimes (q^\ast \mathcal{L})^{-1})\right)^{-1}$$ is an ample line bundle on $X$.

**3. The weak Riemann-Hurwitz formula**

Let $\pi:X\longrightarrow Y$ be a finite morphism of smooth quasi-projective varieties over an algebraically closed field.

Then, the Grothendieck-Riemann-Roch theorem applied to $\pi$ and $\mathcal{O}_X$ gives $$ch(\pi_\ast \mathcal{O}_X) = \pi_\ast( td(X/Y)).$$

In degree 0 this gives something we all know: $c_0(\pi_\ast \mathcal{O}_X)$ is the rank of $\pi_\ast \mathcal{O}_X$ whereas $$\pi_\ast (td(X/Y)_{0} = \pi_\ast (0) = \deg \pi.$$ That is, we get that the rank of the vector bundle $\pi_\ast \mathcal{O}_X$ equals $\deg \pi$.

In degree 1 it gives a weak version of the Riemann-Hurwitz theorem. Namely, it shows that $c_1(\pi_\ast \mathcal{O}_X) = \pi_\ast( td(X/Y)_{(1)})$ in the Chow ring of $Y$ (tensored with $\mathbf{Q}$). I call this version weak because you actually have an equality in the Chow ring of $X$ (tensored with $\mathbf{Q}$).

I should say that this isn't the complete picture yet. The ramification divisor appears when you do a local computation as in Chapter 3.6 Prop. 13 of Serre's book *Local fields*.

In higher degree, you can write out what GRR gives but I can't give a geometric interpretation of this. Maybe someone else can?

**4. Heights for covers of algebraic surfaces in characteristic zero**

Let $k$ be an algebraically closed field of characteristic zero.

Fix an smooth projective connected curve $C$ over $k$ and a flat projective morphism $h:X\longrightarrow C$ with $X$ connected and regular such that the generic fibre $X_\eta$ is nonsingular. Let $D\subset X$ be a simple normal crossings divisor. (This means that its components are nonsingular and meet transversally.)

We now define the set $Cov(C,X,h,D)$ as the set of finite morphisms $\pi:Y \longrightarrow X$ which arise as the normalization of $X$ in the function field of some finite etale morphism $V \longrightarrow X-D$ (with $V$ connected). For any element $\pi:Y \longrightarrow X$ of $Cov(C,X,h,D)$, we have that $\pi$ is finite flat and surjective
and $Y$ is a normal integral complex algebraic surface with rational singularities.

You can define a *height over $C$* on this set and give a nice formula for this height using the Grothendieck-Riemann-Roch theorem. This is all contained in the following

**Theorem.** Let $\pi:Y \longrightarrow X$ be an element of $Cov(C,X,h,D)$. Choose a resolution of singularities $\rho:Y^\prime\longrightarrow Y$ and write $f=h\circ \pi \circ \rho$. Then the first Chern class $c_1(f_! \mathcal{O}_{Y^\prime})$ equals $$f_\ast(td(Y^\prime)_{(2)}) - h_\ast(td(X)_{(1)})td(C)_{(1)} \deg \pi - h_\ast(c_1(\pi_\ast \mathcal{O}_Y))td(C)_{(1)}$$ in the class group of $C$ (tensored with $\mathbf{Q}$). Define the *height over $C$* of $\pi$ to be $$ Height(\pi) = \deg c_1(f_! \mathcal{O}_{Y^\prime}).$$ This height is independent of the resolution $\rho$.

*Proof.*
The formula for $c_1(f_! \mathcal{O}_{Y^\prime})$ is obtained by applying GRR to $(f,\mathcal{O}_{Y^\prime})$ and $(h,\pi_\ast \mathcal{O}_Y)$. The fact that the height is independent of $Y^\prime$ follows from the formula and the Hirzebruch-Riemann-Roch theorem. For details see the proof of Theorem 1.1 in http://arxiv.org/abs/0807.0184 .