For the study of hard ball systems, the algebro-geometric approach
appeared in 1999 --- in a sense surprisingly but quite efficiently
--- for proving the hyperbolicity of typical systems (see
[26]). An improvement by Simányi [22] also provided the
ergodicity of typical systems, thus an almost complete proof of the
Boltzmann--Sinai ergodic hypothesis. More than that, at present, the
best form of the local ergodicity theorem for semi-dispersing
billiards, [6] also uses algebraic methods (and the algebraicity
condition on the scatterers). The goal of the present paper is to
discuss the essential steps of the algebro-geometric approach by
assuming and using possibly minimum information about hard ball
systems. In particular, we also minimize the intersection of the
material with the earlier surveys [29] and [20].