For my research, I'm trying to understand how the Grothendieck--Riemann--Roch theorem is used in the paper The Birational Geometry of the Hilbert Scheme of Points on Surfaces by Aaron Bertram & Izzet Coskun.
Throughout this paper, $X$ is a smooth projective surface over the complex numbers. $X^{[n]}$ denotes the Hilbert scheme parameterizing zero-dimensional schemes of length $n$. $X^{(n)}$ denotes the $n$-th symmetric product of $X$. $B$ is the class of exceptional divisors of Hilbert--Chow morphism.
On page 19, $L$ is assumed to be an ample line bundle on $X$. $L[n]$ is an induced line bundle on $X^{[n]}$. Construction 2 induces a rational map $$\phi_L \colon X^{[n]} \to G(N-n, N).$$ It is written that a straightforward calculation using the Grothendieck--Riemann--Roch theorem shows that the class of $\phi_L^{*}(\mathscr{O}_{G(N-n, N)} (1))$ is $$\phi_L^{*}(\mathscr{O}_{G(N-n, N)} (1)) = L[n] - \frac{B}{2}.$$ I tried to figure out how to apply the GRR theorem but I still do not know.