Hilbert series and applications to graded rings

Selma Altinok

doi:10.1155/s0161171203107090

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.

Highlights

We work with graded rings R = n≥0 Rn that are finitely generated over an algebraically closed field k of characteristic 0 and satisfyR0 = k
If S is a projective surface with Du Val singularities and D a Weil divisor on S, some multiple r D is Cartier, and there is a formula [9, Theorem 9.1]
The point, is that the corollary gives a formula for the Hilbert series PX (t) of the Fano 3-fold in terms of simpler data for a K3 surface

Summary

Introduction

We work with graded rings R = n≥0 Rn that are finitely generated over an algebraically closed field k of characteristic 0 and satisfy R0 = k. The Hilbert function of R is the numerical function Pn = dim Rn for n ≥ 0; the Hilbert series P (t) or PR(t) of R is the formal power series defined by P (t) = Pntn. It is elementary and well known that P (t) is a rational function of t. In fact, if x1, . . . , xd are homogeneous elements of weight wt xi = ai generating R (or more generally, generating a subring over which R is finite), then (1 − tai )P (t) = Q(t) is a polynomial.

Fractional Riemann-Roch formula

Fano 3-folds and K3 surfaces