K3 polytopes and their quartic surfaces

K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to 30 vertices. Their number is 36 297 333. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.

Quantitative Singularity Theory for Random Polynomials

Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-$W$ singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-$W$ singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset $W$ of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-$W$ singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function $p$ we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center $p$, in which the $W$-singularity structure is constant.

Singularity properties of null killing magnetic curves in Minkowski 3-space

We reconstruct the Cartan Equations of null Killing magnetic curve [Formula: see text] in [Formula: see text] with Killing magnetic vector field [Formula: see text] under the new Cartan frame [Formula: see text], which describe some new geometrical properties of [Formula: see text]. The singularity properties of the rectifying surfaces and the binormal osculating surfaces of null Killing magnetic curves are given. As an application, two examples are given to explain the main results, where the singular loci of null Killing magnetic curves are obtained.

Hypersurfaces with linear type singular loci

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ([Formula: see text]) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.

Collections of Hypersurfaces Containing a Curve

We consider the closed locus parameterizing $k$-tuples of hypersurfaces that have positive dimensional intersection and fail to intersect properly, and show in a large range of degrees that its unique irreducible component of maximal dimension consists of tuples of hypersurfaces whose intersection contains a line. We then apply our methods in conjunction with a known reduction to positive characteristic argument to find the unique component of maximal dimension of the locus of hypersurfaces with positive dimensional singular loci. We will also find the components of maximal dimension of the locus of smooth hypersurfaces with a higher dimensional family of lines through a point than expected.