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.



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> Is the sum of a big divisor and a pseudo-effective divisor itself a big divisor?
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Recall that a divisor is called *pseudo-effective* if it lies in the closure of the cone of effective divisors.



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


