On primitive subdivisions of an elementary tetrahedron.
A tetrahedron \(T\subset\mathbb{R}^3\) is called \(L\)-elementary (with respect to some lattice \(L\subset\mathbb{R}^3\)) if \(T\cap L\) exactly consists of the four vertices of \(T\). If addidtionally the volume of~\(T\) equals \(\frac{\det(L)}{6}\), then the tetrahedron is \(L\)-primitive. The main contribution of the paper is a construction that produces for each \(\mathbb{Z}^3\)-elementary tetrahedron \(T\) a subdivision into \(\frac{1}{4}\mathbb{Z}^3\)-primitive tetrahedra (i.e., a set of such tetrahedra, any two of which have a common face as their intersection, and whose union equals~\(T\)). As the construction induces for each bounding triangle \(F\) of~\(T\) a subdivision into the four triangles obtained by joining the three midpoints of the edges of~\(F\), the authors can derive from their construction the main result of the paper: Every convex polytope \(P\subset\mathbb R^3\) with integral vertices admits a subdivision into \(\frac{1}{4}\mathbb{Z}^3\)-primitive tetrahedra. This improves on a corresponding result by \textit{G. Kempf, F. Knudsen, D. Mumford} and \textit{B. Saint-Donat} [Toroidal embeddings. I (1973; Zbl 0271.14017)] guaranteeing a subdivision into \(\tfrac{1}{2^t}\mathbb{Z}^3\)-primitive tetrahedra for some \(t\in\mathbb{N}\) depending on~\(P\).
scientific article; zbMATH DE number 2101103