The complete classification of empty lattice 4-simplices

Blowups with log canonical singularities

“…Empty simplices in dimension 4 were recently classified by Iglesias-Valiño and Santos [9]. Using their classification and the results above we now prove Theorem 1.4, which gives effective bounds on n min for the case of terminal singularities in dimension 4 and answers Question 1.1 completely.…”

“…(In this situation we say p is a generating point of that hollow simplex). This relation makes Theorem 3.1 equivalent to [9,Corollary 2.7].…”

“…Note, however, that the claim made in [2] that the results of [14] and [5] are valid for composite index is incorrect, as was pointed out in [4]. The complete classification was recently given in [9].…”

“…The relation of this with the setting of the previous section is as follows: if G is a closed group containing Z d and with G ∩ ∆ • = ∅ then, as noted in the previous section, π L (∆) is a hollow lattice polytope with respect to the lattice Λ G = π L (G), where L is the identity component of G and π L : R d → R d /L ∼ = R k is the canonical projection. Thus the rational points in G parametrise (perhaps part of) a fine family of hollow simplices in the sense of [9]: each point p ∈ G ∩ Q d corresponds, as in Corollary 2.5, to the standard simplex ∆ ⊂ R d considered with respect to the lattice Λ p = Z d + Zp. (In this situation we say p is a generating point of that hollow simplex).…”

Classification of Triples of Lattice Polytopes with a Given Mixed Volume

“…In the last decade, there has been an increased interest in the algorithmic theory of lattice polytopes, which is motivated by applications in algebra, algebraic geometry, combinatorics, and optimization (see, for example, [3,4,6,[8][9][10][11][12][19][20][21]23]). So far, a special emphasis has been put on computer-assisted enumeration results, which are important from different perspectives.…”