Big real divisor

A big real divisor is an actual positive real combination of big Cartier divisors. Equivalently it is real linearly equivalent, or numerically equivalent, to an ample real divisor plus an effective real divisor. The rational approximation of an ample-plus-effective real divisor and Kodaira's lemma connect these formulations. Fujino's notes on big real divisors give the definition also for nonnormal varieties; bigness under finite normalization relates it to section growth.

Write a big real Cartier divisor on an integral projective variety as $D{\sim }_{\mathbb{R}}B+E$, with $B$ ample and $E$ effective real Cartier. Every curve outside $\mathrm{Supp}E$ has $D\cdot C=B\cdot C+E\cdot C>0$. Therefore curves negative against $D$ lie in finitely many support components of codimension one. On a projective scheme use componentwise bigness on a projective scheme and collect the supports on its finitely many integral components; codimension one is taken in the relevant component.

If a big class $D=B+E$ has ample restriction to every component $E_i$ of the effective support, then for any ample $A$ a sufficiently small $ϵ>0$ makes $D-ϵA$ nef. Use openness of the ample cone to keep $B-ϵA$ and all $(D-ϵA){|}_{E_i}$ ample with one finite minimum of bounds. Curves inside the support use these restrictions; curves outside use effectivity of $E$. Thus one does not need a uniform choice over infinitely many subvarieties.

For a possibly reducible projective scheme, componentwise bigness means that a real Cartier class restricts to a big real divisor on every reduced irreducible component. This specifies the convention needed by positivity arguments that treat all curves. Maximal total section growth alone is weaker: on ${\mathbb{P}}^2⨿{\mathbb{P}}^2$, the bundle $(\mathcal{O}(1),\mathcal{O}(-1))$ has quadratic total growth but negative degree on every line in the second component. Thus its negative curves cannot be confined to finitely many divisors.

Suppose $D{\sim }_{\mathbb{R}}tA+E$ on an integral projective variety, where $t>0$, $A$ is ample Cartier and $E$ is effective real Cartier. Then $D$ is an actual positive combination of big Cartier divisors.

Here is a finite-dimensional proof. First assume $X$ is normal. Express $D$ and $E$ in finite Cartier bases and write $D-tA-E$ as a finite combination of principal Cartier divisors. The union of the supports of these finitely many divisors has finitely many prime components. Their integer multiplicities turn the equality into finitely many rational linear equations and effectivity into finitely many rational linear inequalities. The given coefficient tuple lies in a rational polyhedron. Take its smallest face; within that face it lies in the relative interior, and $t>0$ is an open condition. A small simplex with rational vertices in this relative interior contains the tuple. Each vertex gives $D_{\nu }{\sim }_{\mathbb{Q}}{t}_{\nu }A+E_{\nu }$ with $t_{\nu }>0$ and $E_{\nu }\ge 0$. Clearing denominators and applying Kodaira's lemma shows that $D_{\nu }$ is a positive rational multiple of a big Cartier divisor. Taking the original convex weights proves the required actual equality.

If $X$ is nonnormal, pull the finite Cartier bases and the relation to its finite normalization and impose the same rational equations and effectivity inequalities there. The vertex divisors remain rational Cartier divisors on $X$ because they were constructed in bases from $X$. They are big on the normalization, hence big on $X$ by bigness under finite normalization. This proves the same conclusion. Effectivity is used in the usual effective Cartier sense so that pullback is effective; arbitrary cycles on a nonnormal variety cannot be substituted without defining a compatible divisor theory.

The big cone in $N^1(X{)}_{\mathbb{R}}$ is open and convex: write a big class as ample plus effective, perturb only the ample summand, and use openness of the ample cone. For positive-dimensional $X$, intersection with $H^{n-1}$ for any very ample divisor $H$ is strictly positive on every big class. Hence this cone contains no line and does not contain the zero class.