The complete classification of empty lattice $4$-simplices
Óscar Iglesias Valiño, Francisco Santos
备注：
31 pages, 3 figures, 8 tables. Apart of minor edits the main change in v3 is the inclusion of a supplementary *.txt file with the data for the 2461 sporadic simplices
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.
We give an almost complete classification of empty lattice simplices in dimension 4 using the conjectural results of Mori-Morrison-Morrison, later proved by Sankaran and Bober. In particular, all of these simplices correspond to cyclic quotient singularities, and all but finitely many of them have width bounded by 2.
A lattice $d$-simplex is the convex hull of $d+1$ affinely independent integer points in ${\mathbb R}^d$. It is called empty if it contains no lattice point apart of its $d+1$ vertices. The classification of empty $3$-simplices is known since 1964 (White), based on the fact that they all have width one. But for dimension $4$ no complete classification is known. Haase and Ziegler (2000) enumerated all empty $4$-simplices up to determinant 1000 and based on their results conjectured that after determinant $179$ all empty $4$-simplices have width one or two. We prove this conjecture as follows: -…
The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.
Given integers $k$ and $m$ with $k \geq 2$ and $m \geq 2$, let $P$ be an empty simplex of dimension $(2k-1)$ whose $δ$-polynomial is of the form $1+(m-1)t^k$. In the present paper, the necessary and sufficient condition for the $k$-th dilation $kP$ of $P$ to have a regular unimodular triangulation will be presented.
Consider a class of simplices defined by systems $A x \leq b$ of linear inequalities with $Δ$-modular matrices. A matrix is called $Δ$-modular, if all its rank-order sub-determinants are bounded by $Δ$ in an absolute value. In our work we call a simplex $Δ$-modular, if it can be defined by a system $A x \leq b$ with a $Δ$-modular matrix $A$. And we call a simplex empty, if it contains no points with integer coordinates. In literature, a simplex is called lattice-simplex, if all its vertices have integer coordinates. And a lattice-simplex called empty, if it contains no points with integer coor…
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It is known that a lattice simplex of dimension $d$ corresponds a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$. Conversely, given a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$ such that the sum of all entries of each element is an integer, we can obtain a lattice simplex of dimension $d$. In this paper, we discuss a characterization of Gorenstein simplices in terms of the associated finite abelian groups. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equals $p,p^2$ and $pq$, where $p$ and $q$ are prime numbers …
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