Related papers: Minimal Terracini loci in projective spaces

We study minimally Terracini finite sets of points in the projective plane and we prove that the sequence of the cardinalities of minimally Terracini sets can have any number of gaps for degree great enough.

After a few results on curves, we characterize the smallest nonempty Terracini loci of Veronese and Segre-Veronese varieties. For del Pezzo surfaces, we give a full description of the Terracini loci. Moreover, we present an algorithm to…

We study subsets S of curves X whose double structure does not impose independent conditions to a linear series L, but there are divisors D in |L| singular at all points of S. These subsets form the Terracini loci of X. We investigate…

In this short note we show that, for any ample embedding of a variety of dimension at least two in a projective space, all high enough degree Veronese re-embeddings have non-empty Terracini loci.

We introduce the notion of ample body of a projective variety and use it to prove emptiness results for Terracini loci and specific identifiability results for toric and homogeneous varieties.

We introduce and study properties of the Terracini locus of projective varieties X, which is the locus of finite subsets S of X such that 2S fails to impose independent conditions to a linear system L. Terracini loci are relevant in the…

Let $X_{P}$ be the projective toric surface associated to a lattice polytope $P$. If the number of lattice points lying on the boundary of $P$ is at least $4$, it is known that $X_{P}$ is embeddable into a suitable projective space as zero…

In this short note we develop new methods toward the ultimate goal of classifying geproci sets in $\mathbb P^3$. We apply these methods to show that among sets of $16$ points distributed evenly on $4$ skew lines, up to projective…

We introduce the notion of r-th Terracini locus of a variety and we compute it for at most three points on a Segre variety.

We consider point sets in $\mathbb{Z}_n^2$ where no three points are on a line - also called caps or arcs. For the determination of caps with maximum cardinality and complete caps with minimum cardinality we provide integer linear…

We study special linear systems of surfaces of $\mathbb{P}^3$ interpolating nine points in general position having a quadric as fixed component. By performing degenerations in the blown-up space, we interpret the quadric obstruction in…

We classify all finite linear spaces on at most 15 points admitting a blocking set. There are no such spaces on 11 or fewer points, one on 12 points, one on 13 points, two on 14 points, and five on 15 points. The proof makes extensive use…

We present a complete classification of minimal problems for generic arrangements of points and lines in space observed partially by three calibrated perspective cameras when each line is incident to at most one point. This is a large class…

Using the possibility of computationally determining points on a finite cover of a unirational variety over a finite field, we determine all possibilities for direct Gorenstein linkages between general sets of points in P^3 over an…

We present a proof of the Harbourne-Hirschowitz conjecture for linear systems with base points of multiplicity seven or less. This proof uses a well-known degeneration of the projective plane, as well as a combinatorial technique that…

We study Waring rank decompositions for cubic forms of rank $n+2$ in $n+1$ variables. In this setting, we prove that if a concise form has more than one non-redundant decomposition of length $n+2$, then all such decompositions share at…

We consider three dimensional piecewise linear cones in $\mathbb{R}^4$ that are mass minimizing w.r.t. Lipschitz maps in the sense of \cite{almgren1976existence} as in \cite{Taylor76}. There are three that arise naturally by taking products…

The Terracini locus $\mathbb{T}(n, d; x)$ is the locus of all finite subsets $S$ of $ \mathbb{P}^n$ of cardinality $x$ such that $\langle S \rangle = \mathbb{P}^n$, $h^0(\mathcal{I}_{2S}(d)) > 0$, and $h^1(\mathcal{I}_{2S}(d)) > 0$. The…

Given a set $P$ of $n$ points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other. We show that the minimum number of lines needed to separate $n$ points, picked randomly…

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…