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.
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.
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.
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.