This paper treats a small but significant error in the model-theoretic
literature on unimodularity.
A unimodular strongly minimal set $D$ is one in which for any interalgebraic
independent tuples \atup and \btup, we have equality of multiplicities
mult(\atup/\btup)=mult(\btup/\atup). Unimodularity was defined and studied in
[hrushovski-unimod], generalising Zilber's proof of local modularity of
locally finite (e.g. \omega-categorical) strongly minimal sets by proving that
unimodularity implies local modularity.
In the abstract and introduction of [hrushovski-unimod], it is indicated that
we can take as an alternative definition: if f_1 and f_2 are definable
surjective functions U \surjects V which are everywhere k_1-to-1 resp.
k_2-to-1, then k_1 = k_2.
The paper under review terms this ``weak unimodularity''. If ``definable'' is
replaced with ``type-definable'' in its definition, it does indeed become
equivalent to unimodularity. This is implicit in the discussion at the start
of section 2 of [hrushovski-unimod], and is spelt out in the reviewed paper.
As stated, however, weak modularity is indeed strictly weaker than
unimodularity - the reviewed paper proves this by giving a simple
counterexample.
This error of assuming equivalence of unimodularity and weak unimodularity has
meanwhile propogated through the literature on measurable structures in the
sense of Macpherson and Steinhorn [MS-1dimAsymp]. [MS-1dimAsymp] and
[Elwes-asymp] each conflate weak unimodularity with unimodularity, and the
latter gives an erroneous proof of the equivalence.
The reviewed paper clarifies resulting issues. The authors note that the
Zilber functions defined in [hrushovski-unimod] correspond to
Macpherson-Steinhorn-measures in the strongly minimal case, showing that for
strongly minimal sets measurablity is equivalent to unimodularity. They
further give a correct proof of the result in [Elwes-asymp] that
Macpherson-Steinhorn-measurable stable theories are 1-based - using the
Buechler dichotomy to reduce to the strongly minimal case and then applying
Hrushovski's theorem that unimodular strongly minimal sets are locally
modular.
The paper is clear and precise.