Volume 19, Issue 4

The present paper generalizes, inside the class of projective toric varieties, the classification [2], performed by Batyrev in 1991 for smooth complete toric varieties, to the singular ℚ-factorial case.

For a fixed hyperbolic quadric 𝓗 in PG(3, q ), let 𝔼 (respectively 𝕋, 𝕊) denote the set of all lines of PG(3, q ) which are external (respectively tangent, secant) with respect to 𝓗. We characterize the minimum size blocking sets of PG(3, q ) with respect to each of the line sets 𝕊, 𝕋 ∪ 𝕊 and 𝔼 ∪ 𝕊.

Several well known polytopal constructions are examined from the functorial point of view. A naive analogy between the Billera–Sturmfels fiber polytope and the abelian kernel is disproved by an infinite explicit series of polytopes. A correct functorial formula is provided in terms of an affine-compact substitute of the abelian kernel. The dual cokernel object is almost always the natural affine projection. The Mond–Smith–van Straten space of sandwiched simplices, useful in stochastic factorizations, leads to a different kind of affine-compact functors and new challenges in polytope theory.

Let 𝓜 g be the moduli space of compact connected hyperbolic surfaces of genus g ≥ 2, and 𝓑 g ⊂ 𝓜 g its branch locus. Let Mg^$\begin{array}{} \widehat{{\mathcal{M}}_{g}} \end{array} $ be the Deligne–Mumford compactification of the moduli space of smooth, complete, connected surfaces of genus g ≥ 2 over ℂ. The branch locus 𝓑 g is stratified by smooth locally closed equisymmetric strata, where a stratum consists of hyperbolic surfaces with equivalent action of their orientation-preserving isometry group. Any stratum can be determined by a certain epimorphism Φ . In this paper, for any of these strata, we describe the topological type of its limits points in 𝓜͡ g in terms of Φ . We apply our method to the 2-complex dimensional stratum corresponding to the pyramidal hyperbolic surfaces.

Let P 1 , …, P n and Q 1 , …, Q n be convex polytopes in ℝ n with P i ⊆ Q i . It is well-known that the mixed volume is monotone: V ( P 1 , …, P n ) ≤ V ( Q 1 , …, Q n ). We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P 1 , …, P n whose number of isolated solutions equals the normalized volume of the convex hull of P 1 ∪ … ∪ P n . In addition, we obtain an analog of Cramer’s rule for sparse polynomial systems.

We give criteria for which a principal curvature becomes a bounded C ∞ -function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.

To study the splitting of nodal plane curves with respect to contact conics, we define the splitting type of such curves and show that it can be used as an invariant to distinguish the embedded topology of plane curves. We also give a criterion to determine the splitting type in terms of the configuration of the nodes and tangent points. As an application, we construct sextics and contact conics with prescribed splitting types, which give rise to new Zariski-triples.

In this paper we correct a mistake in [C. Ciliberto, F. Russo, On surfaces with two apparent double points. Adv. Geom . 5 (2005), 223–235].