= Bigness under finite normalization
Let $\nu:X^\nu\to X$ be the finite normalization of an integral projective variety. The <coherent sheaf> $\mathcal Q=\nu_*\mathcal O_{X^\nu}/\mathcal O_X$ is supported in dimension at most $n-1$. For a <Cartier divisor> $D$, the <projection formula for sheaves> and the resulting <long exact sequence in sheaf cohomology> give
$$
0\leq h^0(X^\nu,m\nu^*D)-h^0(X,mD)\leq h^0(X,\mathcal Q\otimes\mathcal O_X(mD))=O(m^{n-1}).
$$
The last step uses the <polynomial bound for sections of a fixed divisor>. Therefore the leading order $m^n$ growth, and hence bigness, is preserved in both directions. This handles nonnormal varieties without assuming a resolution of singularities in positive characteristic.
Back to article page
