Applications of Grothendieck-Riemann-Roch?

Check out Harris & Morrison's "Moduli of Curves", section 3E. There is a wealth of examples of applications of GRR coming from moduli theory, in which one applies it to projection from the universal family or some fibered power of the universal family. The basic idea in these cases is that both the base space and the total space are rather complicated beasts but the fibers of the morphisms are usually quite tractable, since they are just the gadgets you are trying to parametrize.

For more examples in the same vein, you could read the classic "Towards an enumerative geometry of the moduli space of curves" by David Mumford.

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 .

Whenever you have a Fourier-Mukai transorm, if you want to compute the Chern character of an image of a sheaf, you need GRR.I think you can find examples in the Huybrecht's book "Fourier-Muaki transforms in algebraic geometry".