Classification of smooth weak Fano three-folds A variety is said to be 'weak Fano' or 'almost Fano' if its anti-canonical divisor is nef and big. In this question, let me restrict to the case of smooth varieties over the complex numbers.
My question is: In the case of three-folds, is it known how many deformation families there are of smooth weak Fano varieties?
In the case of two-folds, the number of deformation families of smooth weak Fano varieties is well-known and small - see for example this paper. In the case of three-folds, for genuinely Fano three-folds it's known that there are 105 deformation families, whose properties are collected here. But I haven't been able to find a statement about the general weak Fano case.
 A: I would say it is almost certainly open.
Here is a paper from 2010 which classifies smooth weak Fano $3$-folds with Picard rank 2 satisfying a certain condition on their extremal rays. https://arxiv.org/abs/1009.5036. This strongly suggests a complete classification is not available, now. I suggest looking at this paper, and  papers that reference it ( MathSciNet is usually the best way to do that).
Note that smooth toric weakened Fano 3-folds have been classified by Sato, and there are 15 classes in addition to toric Fano 3-folds. https://arxiv.org/pdf/math/0201258.pdf. All of the examples are the ones which one would think to write down, i.e. weak del Pezzo surface bundles over $\mathbb{P}^1$ and $\mathbb{P}^1$-bundles over weak del Pezzo surfaces.
Here weakend means that $-K_{X}$ is big and nef and $X$ deforms to a Fano variety.
Note however that weakened $\implies$ weak but the converse is false.
As far as I can tell there is no complete classifiation of smooth toric weak fano 3-folds at this time, which one would expect to come long before a classification in the general case.
