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 seriesP(t)in a number of cases of interest for singular surfaces (see Lemma 2.1) and3-folds. IfXis aℚ-Fano3-fold andS∈ |−KX|aK3surface in its anticanonical system (or the general elephant ofX), polarised withD=𝒪S (−KX), we determine the relation betweenPX(t)andPS,D(t). We discuss the denominator∏(1−tai)ofP(t)and, in particular, the question of how to choose a reasonably small denominator. This idea has applications to findingK3surfaces and Fano3-folds whose corresponding graded rings have small codimension. Most of the information about the anticanonical ring of a Fano3-fold orK3surface is contained in its Hilbert series. We believe that, by using information on Hilbert series, the classification ofℚ-Fano3-folds is too close. FindingK3surfaces are important because they occur as the general elephant of aℚ-Fano 3-fold.