Negative deformations of toric singularities that are smooth in codimension two

Given a polyhedral cone sigma with smooth two-dimensional faces and, moreover, a lattice point R in the dual cone of sigma, we describe the part of the versal deformation of the associated toric variety TV(sigma) that is built from the deformation parameters of multidegree R. Let Q the polyhedron obtained by intersecting sigma with the hyperplane R=1. Then the base space is (the germ of) an affine scheme that reflects certain possibilities of splitting Q into Minkowski summands

The Newton polytope of the discriminant of a quaternary cubic form

We determine the 166104 extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with D-equivalence classes of regular triangulations of the 3-dilated tetrahedron. We describe how to compute these triangulations and their D-equivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer number of triangulations effectively, as well as devising a suitably fast algorithm for computation of a D-equivalence class

Toric Geometry in OSCAR

We report on the computer implementation for toric geometry in the computer algebra system $\texttt{OSCAR}$. The main architectural feature of $\texttt{OSCAR}$ is that its four fundamental tools $\texttt{Antic}$ (Hecke, Nemo), $\texttt{GAP}$, $\texttt{Polymake}$ and $\texttt{Singular}$ are $\mathit{integral~components}$, rather than external software. Toric geometry benefits greatly from this architecture. $\texttt{Julia}$ is a high-performance programming language designed for numerical and scientific computing. The growing ecosystem of $\texttt{Julia}$ packages ensures its continued viability for scientific computing and data analysis. Indeed, $\texttt{OSCAR}$ is written in $\texttt{Julia}$. This implies that the performance of $\texttt{OSCAR}$ should be comparable or even better than many other implementations.Comment: 6 pages, prepared for the ComputerAlgebraRundbrief (March 2023

NGC 6302: The Tempestuous Life of a Butterfly

NGC 6302 (The ''Butterfly Nebula'') is an extremely energetic bipolar nebula whose central star is among the most massive, hottest, and presumably rapidly evolving of all central stars of planetary nebulae. Our proper-motion study of NGC 6302, based on excellent HST WFC3 images spanning 11 yr, has uncovered at least four different pairs of expanding internal lobes that were ejected at various times over the past two millennia at speeds ranging from 10 to 600 km s^-1. In addition, we find a pair of off-axis flows in constant motion at 760 +/- 100 km s^-1 within which bright [Fe II] feathers are conspicuous. Combining our results with those previously published, we find that the ensemble of flows has an ionized mass > 0.1 M_sun. The kinetic energy of the ensemble, 10^46 - 10^48 ergs, lies at the upper end of gravity-powered processes such as stellar mergers or mass accretion and is too large to be explained by stellar radiation pressure or convective ejections. The structure and dynamics of the Butterfly Nebula suggests that its central engine has had a remarkable history, and the highly unusual patterns of growth within its wings challenge our current understanding of late stellar mass ejection.Comment: 17 pages, 3 figures, 1 tabl