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
∏(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.