Parallel Enumeration of Triangulations

New Counts for the Number of Triangulations of Cyclic Polytopes

“…Here we address the problem raised in [10, §5.2], which asks for determining the number of triangulations of C(n, d). We report on new computational results, obtained via the new software MPTOPCOM [9]. This verifies and extends previous results of Rambau and Reiner [13, ∆] is a down-flip.…”

K3 Polytopes and their Quartic Surfaces

“…The moduli space of quartic surfaces in P 3 is the projective variety determined by this invariant ring, namely Proj C[HS 4,3 ] SL (4) . Following Mumford [20,21], we give the following definitions.…”

“…We use the notation HS s 4,3 and HS ss 4,3 to denote the set of stable and semistable points respectively. The GIT quotient of the action of SL( 4) is defined on the semistable locus HS ss 4,3 , as follows: ϕ : HS ss 4,3 → HS 4,3 //SL(4) := Proj C[HS 4,3 ] SL (4) . The image ϕ(HS s 4,3 ) of the stable locus is the moduli space of stable quartic surfaces in P 3 .…”

“…If α = 0 then we argue as in the proof of [8,Theorem 2.51] and conclude that the type is D 5 . If α = 0, by running through our list of polynomials, we find that they all contain either yz 3 or yz 4 . In the first situation, after applying the Tschirnhaus transformation described in the proof of the aforementioned theorem, we get a singularity of type D 6 :…”

“…They use different algorithms to pass through cones of the secondary fan: gfan computes a new weight by traversing a facet, while TOPCOM exploits bistellar flips. Jordan, Joswig and Kastner [4] introduced a new algorithm, called down-flip reverse search, for parallel enumeration of regular triangulations. Their implementation mptopcom generated results that are out of reach for gfan and TOPCOM.…”