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Let $\nu :X^{\nu }\to X$ be the finite normalization of an integral projective variety. The coherent sheaf $\mathcal{Q}={\nu }_{\ast }{\mathcal{O}}_{X^{\nu }}/{\mathcal{O}}_X$ 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 giveThe 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.

Let $N$ be a nef Cartier divisor and let the coherent sheaf $\mathcal{F}$ have support dimension $d$. Then $h^i(X,\mathcal{F}(mN))=O(m^{d-i})$ for $1\le i\le d$. This does not require reducedness or smoothness.

Choose a very ample $A$ and fix $a$ sufficiently large for Fujita vanishing. Choose a section of $aA$ avoiding the associated points of $\mathcal{F}$. Its multiplication gives $0\to \mathcal{F}\to \mathcal{F}(aA)\to \mathcal{Q}\to 0$, where $\mathcal{Q}$ has support dimension at most $d-1$. After tensoring with ${\mathcal{O}}_X(mN)$, the middle term has zero higher sheaf cohomology, uniformly in $m$. Hence $h^i(\mathcal{F}(mN))\le h^{i-1}(\mathcal{Q}(mN))$. Induct on support dimension: for $i=1$ use the polynomial bound for sections of a fixed divisor, and for $i>1$ use the induction hypothesis. Degree greater than $d$ vanishes by Grothendieck vanishing.

For an effective Cartier divisor $F$, the divisor restriction exact sequence givesThe scheme $F$ has dimension at most $n-1$, so the permitted scheme version of part (i) bounds $h^0(F,{\mathcal{O}}_F(kD))$ by $C{k}^{n-1}$. For the infinitely many $k$ in the hypothesis,once $k$ is sufficiently large. Thus infinitely many such $k$ have a nonzero section of $kD-F$. This dimension-drop argument is the section subtraction lemma for big divisors.

If “effective divisor” is interpreted as an effective Weil divisor on a normal variety, use its coherent divisor ideal instead. The quotient by that ideal is supported in dimension at most $n-1$, so the polynomial bound for sections of a fixed divisor gives the same conclusion. For $F=0$ the assertion is immediate.

Choose a very ample divisor $A$ such that $A-D$ has a nonzero global section; this is possible by taking a sufficiently high ample twist. Since $X$ is integral, multiplication by its $m$th power injects $H^0(X,mD)$ into $H^0(X,mA)$. By Serre vanishing and the Hilbert polynomial, $h^0(X,mA)=O(m^n)$. ConsequentlyThis is the polynomial bound for sections of a fixed divisor. The same proof works for a coherent sheaf of support dimension $d$, by choosing the multiplying section to avoid its associated points; the bound is then $O(m^d)$.

If $D$ is a big divisor and $F$ an effective Cartier divisor, infinitely many $m$ have $H^0(X,mD-F)\ne 0$. The divisor restriction exact sequence bounds the dimension lost upon restriction to $F$ by $O(m^{n-1})$, using the polynomial bound for sections of a fixed divisor; this cannot exhaust the $c{m}^n$ sections along the infinite growth sequence.