The classification of 4-dimensional empty simplices
Abstract: Empty simplices, that is, lattice simplices with no lattice points except their vertices, are interesting from at least two points of view. On the one hand, every lattice polytope can be triangulated into empty simplices. On the other hand, empty simplices correspond to the terminal quotient singularities of the minimal model program. In dimension two the only empty simplex is the unimodular triangle, and in dimension three the classification is known since 1964 (White) and sometimes called the “terminal lemma” by algebraic geometers. The main feature of it is that every empty tetrahedron has “width one” with respect to some integer linear functional, which easily implies that there is a classification of empty tetrahedra by two integer parameters, one of which is its determinant or “normalized volume”.
In this talk I report on a complete classification of four dimensional empty simplices, which goes as follows.
– Empty simplices of width one, which form a 3-parameter family.
– Empty simplices that admit an integer projection to the second dilation of a unimodular triangle, which form two 2-parameter families.
– Empty simplices that admit an integer projection to a triangular bipyramid without interior lattice points, which form 52 1-parameter families.
– Empty simplices that do not project to any 3-dimensional lattice polytope without interior points, of which there are exactly 2461, with determinants ranging from 24 to 419.
Our methods combine a computer enumeration of empty simplices of volume up to 7600 with discrete-geometric and geometry-of-numbers techniques to bound the volume of lattice simplices with certain properties. Our classification corrects statements made by Barile, Bernardi, Borisov and Kantor (2011), who missed about half of the 1-parameter and 2-parameter families. It also corrects the computational output of Mori, Morrison, and Morrison (1988), who computed all empty 4-simplices of prime volume smaller than 1600.
This is joint work with Oscar Iglesias, using also joint results with Christian Haase, Jan Hofmann, and Mónica Blanco.