International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 7, Pages 397-403

Hilbert series and applications to graded rings

Selma Altinok

Department of Mathematics, Arts and Science Faculty, Adnan Menderes University, Aydın 09010, Turkey

Received 23 July 2001; Revised 14 March 2002

Copyright © 2003 Selma Altinok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 -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 (1tai) 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 -Fano 3-folds is too close. Finding K3 surfaces are important because they occur as the general elephant of a -Fano 3-fold.