Kawamata boundedness for Fano threefolds and the Graded Ring Database

We explain an effective Kawamata boundedness result for Mori–Fano 3-folds. In particular, we describe a list of 39,550 possible Hilbert series of semistable Mori–Fano 3-folds, with examples to explain its meaning, its relationship to known classifications and the wealth of more general Fano 3-folds it contains, as well as its application to the on-going classification of Fano 3-folds.

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