Asymptotic Riemann–Roch

For a Cartier divisor $D$ on an $n$-dimensional projective scheme, $\chi (X,{\mathcal{O}}_X(mD))$ is a polynomial in $m$ whose leading term is $D^n{m}^n/n!$. The intersection product uses the fundamental cycle, with the generic multiplicity of each component. Thus the statement also applies to nonreduced schemes.