  An empty simplex is a lattice simplex with only its vertices as lattice
points. Their classification in dimension three was completed by White in 1964.
In dimension four, the same task was started in 1988 by Mori, Morrison, and
Morrison, with their motivation coming from the close relationship between
empty simplices and terminal quotient singularities. They conjectured a
classification of empty simplices of prime volume, modulo finitely many
exceptions. Their conjecture was proved by Sankaran (1990) with a simplified
proof by Bober (2009). The same classification was claimed by Barile et al. in
2011 for simplices of non-prime volume, but this statement was proved wrong by
Blanco et al. (2016+).
  In this article we complete the classification of $4$-dimensional empty
simplices. In doing so we correct and complete the classification claimed by
Barile et al., and we also compute all the finitely many exceptions, by first
proving an upper bound for their volume. The whole classification has:
  - One $3$-parameter family, consisting of simplices of width equal to one.
  - Two $2$-parameter families (the one in Mori et al., plus a second new one).
  - Forty-six $1$-parameter families (the 29 in Mori et al., plus 17 new ones).
  - $2461$ individual simplices not belonging to the above families, with
volumes ranging between 29 and 419.
  We characterize the infinite families of empty simplices in terms of lower
dimensional point configurations that they project to, with techniques that can
be applied to higher dimensions and larger classes of lattice polytopes.

Combinatorics;Algebraic Geometry