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∈ |−KX| a K3 surface in its anticanonical system (or the general elephant of X), polarised with D=𝒪S (−KX), we determine the relation between PX(t) and PS,D(t). We discuss the denominator ∏(1−tai) of P(t) and, in particular, the question of how to choose a reasonably small denominator. This idea has applications to finding K3 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 K3 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 K3 surfaces are important because they occur as the general elephant of a ℚ-Fano 3-fold.