Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 139 / 1

For a projective scheme $X$ of dimension of a scheme $n$ and a Cartier divisor $D$, asymptotic Riemann–Roch givesHere Euler characteristic of a coherent sheaf means $\chi ={\sum }_{i=0}^n(-1{)}^i{h}^i$, and $D^n$ is the degree of the top intersection product with the fundamental cycle of $X$, including its component multiplicities. No ampleness assumption on $D$ is needed.

The Nakai–Moishezon criterion says that a Cartier divisor $D$ on a projective scheme is ample exactly whenfor every positive-dimensional integral closed subvariety $Y$. Kleiman's criterion says that the ample cone is the interior of the nef cone; equivalently, the numerical class of $D$ is ample exactly when it is strictly positive on every nonzero element of the closed cone of curves $\overline{\mathrm{NE}}(X)$. Positivity merely on individual curves is insufficient: the closure of the cone is essential. A nef divisor has nonnegative intersection number with every integral curve, and its restriction to every closed subscheme is nef.

Choose very ample divisors $H_2$ and $H_1\sim D+H_2$, with defining sections avoiding the associated points of $X$. Such a choice is possible after taking a sufficiently high ample twist. Put $R_m=mD-H_1=(m-1)D-H_2$. The two indicated short exact sequences of sheaves and their long exact sequences in sheaf cohomology implyWe abbreviate $h^i(Z,{\mathcal{O}}_Z(L))$ by $h^i(Z,L)$. Each $H_a$ has dimension $n-1$ and the restricted divisor is nef. For positive degrees below $n-1$, the assumed induction estimate bounds the error terms by $O(m^{n-2})$. The displayed hypothesis omits top-degree cohomology; use top cohomology boundedness for nef twists for that degree, and Grothendieck vanishing above it. These give the same error bound. The auxiliary top-degree result follows from Fujita vanishing by cutting a coherent sheaf with an ample divisor, as proved in cohomology growth for nef twists.

For $n\ge 2$, summing the inequality over successive $m$ gives $h^j(X,mD)=O(m^{n-1})$ for every $j>1$. For $n\le 1$, all such groups vanish by Grothendieck vanishing.

By asymptotic Riemann–Roch and the preceding bounds for higher sheaf cohomology,A nef divisor satisfies $D^n\ge 0$: for a fixed ample divisor $A$, the divisors $D+\epsilon A$ are ample for positive rational $\epsilon$, and continuity of the intersection product gives the inequality as $\epsilon \to 0$. If $h^0(X,mD)=0$ eventually, the left side is nonpositive. Dividing by $m^n$ forces $D^n=0$, since $h^1\ge 0$ and $D^n\ge 0$. The same identity then gives $h^1(X,mD)=O(m^{n-1})$.

First suppose $X$ is integral. Induct on its dimension of a scheme. The previous two parts handle all $j>1$ and also handle $j=1$ when $h^0(X,mD)$ eventually vanishes. Otherwise choose $r>0$ with a nonzero global section of $rD$. On an integral variety it defines an effective Cartier divisor $E$, possibly empty, and multiplication by the section givesThus $h^1(X,mD)\le h^1(X,(m-r)D)+h^1(E,mD)$. For $n\ge 2$, the induction hypothesis bounds the last term by $O(m^{n-2})$; summing on each residue class modulo $r$ gives $O(m^{n-1})$. For $n=1$, $E$ is zero-dimensional and its positive-degree sheaf cohomology vanishes, so the same recurrence is bounded.

For a general projective scheme, a nonzero section can be a zero divisor, so that argument requires an additional step. The general cohomology growth for nef twists supplies it: for every coherent sheaf $\mathcal{F}$ with support dimension $d$,Its proof uses Fujita vanishing, an ample section avoiding the associated points of $\mathcal{F}$, and induction on support dimension. Taking $\mathcal{F}={\mathcal{O}}_X$ gives the required $O(m^{n-1})$ estimate for every $j\ge 1$, including nonreduced and reducible schemes; degrees above $n$ vanish.

The divisor is understood to be on $X$; the printed “on $D$” is a typo. Combining asymptotic Riemann–Roch with cohomology growth for nef twists givesSince a nef divisor has nonnegative top self-intersection number, this limit is positive exactly when $D^n>0$. This is the volume of a nef divisor criterion.