Hilbert series and applications to graded rings

Abstract

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 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.