# Sum of a big divisor and a pseudo-effective divisor

Let $X$ be a smooth projective variety over a field $k$.

It is well-known that the sum of an ample divisor and an effective divisor is big (in fact this can be taken to be the definition of a big divisor). I'm looking for a weakening of this.

Is the sum of a big divisor and a pseudo-effective divisor itself a big divisor?

Recall that a divisor is called pseudo-effective if it lies in the closure of the cone of effective divisors.

Let's say the big divisor is $B$ and the pseudoeffective one is $F$. Write $B = A + E$ with $A$ ample and $E$ effective (by Kodaira). Then $B+F = A+E+F = A/2 + E + (F + A/2)$. $F+A/2$ is effective since $F$ is psef, and so we've written $B+F$ as ample plus effective, so it's big.
• This looks promising. Though why is $F + A/2$ effective? – Daniel Loughran Jun 27 '17 at 16:01
• @DanielLoughran All classes in some ball around $A/2$ are ample, hence effective. One can look at a ball around $F$ of the same size, and find an effective class in it because $F$ is pseudoeffective. Then $F + A/2$ is that class plus an ample class. – Will Sawin Jun 27 '17 at 20:58