Total equimodularity: simplicial cones, Hilbert basis, and triangulation

Related topics: grassmanians, affine toric geometry, desingularization in toric geometry
Tools used: linear algebra, easy-peasy number theory, Gröbner basis and regular triangulations (Sturmfels), toric ideals, more linear algebra, combinatorics of polytopes

After a short introduction to combinatorial optimization, polyhedral geometry and integer decomposition properties, we will focus on totally equimodular simplicial cones.
Here are some quick definitions.
A
simplicial cone is the positive span of linearly independent vectors of $\mathbb{R}^n$.
It is
rational, if it is generated by rational, or equivalently integer, vectors.
The
Hilbert basis of a rational cone is the inclusionwise minimal set of integer points which satisfies: "all integer points of the cone can be expressed as a nonnegative integer combination of the Hilbert basis".
A cone generated by a $\mathbb{Z}$-basis $B$ of $\mathbb{Z}^n$ is called
unimodular and has a trivial Hilbert basis which is $B$.
A cone is totally equimodular if taking any subset of its generators, the associated matrix has all its maximum size determinants of same absolute value.
A
Hilbert triangulation of a cone $C$ is a collection of cones in $C$, whose generators are Hilbert basis elements, and whose union is $C$, and which pairwise do not intersect in their interior.
Using a theorem of Chervet and Grappe (private communication), we find the Hilbert basis of every totally equimodular simplicial cones and many unimodular Hilbert triangulations in (nearly) every cases.