Hilbert series and applications to graded rings

This paper contains a number of practical remarks on Hilbert series that we expect to be useful in various contexts. We use the fractional Riemann-Roch formula of Fletcher and Reid to write out explicit formulas for the Hilbert series P ( t ) in a number of cases of interest for singular surfaces (see Lemma 2.1) and 3 -folds. If X is a ℚ -Fano 3 -fold and S ∈ | − K X | a K 3 surface in its anticanonical system (or the general elephant of X ), polarised with D = 𝒪 S ( − K X ) , we determine the relation between P X ( t ) and P S , D ( t ) . We discuss the denominator ∏ ( 1 − t a i ) of P ( t ) and, in particular, the question of how to choose a reasonably small denominator. This idea has applications to finding K 3 surfaces and Fano 3 -folds whose corresponding graded rings have small codimension. Most of the information about the anticanonical ring of a Fano 3 -fold or K 3 surface is contained in its Hilbert series. We believe that, by using information on Hilbert series, the classification of ℚ -Fano 3 -folds is too close. Finding K 3 surfaces are important because they occur as the general elephant of a ℚ -Fano 3-fold.