Related papers: Minimal plus-one generated line arrangements with …
We propose a new definition of the elliptic genera for complete intersections, not necessarily nonsingular, in projective spaces. We also prove they coincide with the expressions obtained from Landau-Ginzburg model by an elementary…
Let ${\mathcal G}_k(V)$ be the $k$-Grassmannian of a vector space $V$ with $\dim V=n$. Given a hyperplane $H$ of ${\mathcal G}_k(V)$, we define in [I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating $k$-linear forms, J.…
Multinets are certain configurations of lines and points with multiplicities in the complex projective plane P2. They are used in the studies of resonance and characteristic varieties of complex hyperplane arrangement complements and…
In 1933 von Neumann proved a beautiful result that one can approximate a point in the intersection of two convex sets by alternating projections, i.e., successively projecting on one set and then the other. This algorithm assumes that one…
We develop a constructive process which determines all extreme points of the unit ball of the space of $m$--linear forms, $m\geq1.$ Our method provides a full characterization of the geometry of that space through finitely many elementary…
We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.
We classify the unimodular equivalence classes of inclusion-minimal polygons with a certain fixed lattice width. As a corollary, we find a sharp upper bound on the number of lattice points of these minimal polygons.
We prove tight upper bounds for the number of vertices of a simple polygon that is the union or the intersection of two simple polygons with given numbers of convex and concave vertices. The similar question on graphs of the lower (or…
We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.
A compact complex surface with positive definite intersection lattice is either the projective plane or a false projective plane. If the intersection lattice is negative definite, the surface is either a non-minimal secondary Kodaira…
We give a complete combinatorial characterization of weakly $d$-Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any sufficiently large general position point set in $\mathbb…
We study sets of triple points of B\"{o}r\"{o}czky's arrangements of lines in the context of the containment problem proposed by Harbourne and Huneke. We show that in the class of those arrangements, the smallest counterexample to the…
Under suitable technical assumptions, a description is given for the generators of $s$-residual intersections of an ideal $I$ in terms of lower residual intersections, if $s \geq \mu(I)-2$. This implies that $s$-residual intersections can…
The main goal of this note is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erd\H{o}s-type inequality and Hirzebruch-type inequality for a certain class of conic-line…
Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete…
This brief note gives a survey on results relating to existence of closed points on schemes, including an elementary topological characterization of the schemes with (at least one) closed point.
We present a method to find globally optimal topology and trajectory jointly for planar linkages. Planar linkage structures can generate complex end-effector trajectories using only a single rotational actuator, which is very useful in…
In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement with at most triple points, we…
Let ${\mathcal C}(\Omega)$ be the linear code arising from a projective system $\Omega$ of $\mathrm{PG}(V).$ Consider the point-line geometry $\Gamma=({\mathcal P},{\mathcal L})$ and a projective embedding $\varepsilon\colon…
One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erd\H{o}s: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a…