Hodge numbers and deformations of Fano 3-folds

We show that index 1 Fano 3-folds which lie in weighted Grassmannians in their total anticanonical embedding have finite automorphism group, and we relate the deformation theory of any Fano 3-fold that has a K3 elephant to its Hodge theory. Combining these results with standard Gorenstein projection techniques calculates both the number of deformations and the Hodge numbers of most quasismooth Fano 3-folds in low codimension. This provides detailed new information for hundreds of families of Fano 3-folds.