Related papers: Minimal decomposition of binary forms with respect…

It was recently determined exactly through how many general points a nondegenerate curve with nonspecial hyperplane section can pass. This gives rise to a method of constructing reducible curves $C_1 \cup_\Gamma C_2 \to \mathbb{P}^r$ with…

For an irreducible variety $X$ over a field $k$, the degree of irrationality $\operatorname{irr}_k X$ is the minimal degree of a dominant rational map $X \dashrightarrow \mathbb{P}_k^{\operatorname{\dim} X}$. When $X$ is a curve, this is…

For each of n=1,2,3 we find the minimal height h^(P) of a nontorsion point P of an elliptic curve E over C(T) of discriminant degree d=12n (equivalently, of arithmetic genus n), and exhibit all (E,P) attaining this minimum. The minimal…

A minimal separating set in a connected topological space $X$ is a subset $L \subset X$ with the property that $X \setminus L$ is disconnected, but if $L^{\prime}$ is a proper subset of $L$, then $X \setminus L^{\prime}$ is connected. Such…

A rational map with good reduction in the field $\mathbb{Q}\_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\mathbb{P}^1(\mathbb{Q}\_p)$ over $\mathbb{Q}\_p$. The dynamical structure of such a system…

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness…

In the realm of tensor optimization, the low-rank Tucker decomposition is crucial for reducing the number of parameters and for saving storage. We explore the geometry of Tucker tensor varieties -- the set of tensors with bounded Tucker…

Let $X\subset P^n$ be a complex projective manifold of degree $d$ and arbitrary dimension. The main result of this paper gives a classification of such manifolds (assumed moreover to be connected, non-degenerate and linearly normal) in case…

Tangent and normal cones play an important role in constrained optimization to describe admissible search directions and, in particular, to formulate optimality conditions. They notably appear in various recent algorithms for both smooth…

Let $S$ and $T$ be reduced divisors on $\mathbb{P}^2$ which have no common components, and $\Delta=S+2\,T.$ We assume $\deg\Delta=6.$ Let $\pi:X\to\mathbb{P}^2$ be a normal triple cover with branch divisor $\Delta,$ i.e. $\pi$ is ramified…

Determinantal varieties -- the sets of bounded-rank matrices or tensors -- have attracted growing interest in low-rank optimization. The tangent cone to low-rank sets is widely studied and underpins a range of geometric methods. The…

We classify the simple linear compactifications of SO(2r+1), namely those compactifications with a unique closed orbit which are obtained by taking the closure of the SO(2r+1)xSO(2r+1)-orbit of the identity in a projective space P(End(V)),…

Let $(X,o)$ be a germ of a 3-dimensional terminal singularity of index $m\geq 2$. If $(X,o)$ has type cAx/4, cD/3-3, cD/2-2, or cE/2, then assume that the standard equation of $X$ in $\mathbb{C}^4/\mathbb{Z}_m$ is non-degenerate with…

The notion of 'slope rational connectedness' is introduced in the context of smooth orbifold pairs. The main result parallels the characterization of the rational connectedness of projective manifolds in terms of either the non-existence of…

This is another proof of the same result in [9]. Let $X_0$ be a generic quintic hypersurface in $\mathbf P^4$ over $\mathbb C$ and $c_0$ a regular map $\mathbf P^1\to X_0$ that is generically one-to-one to its image. In this paper, we show…

The minimum rank of a graph G is the minimum rank over all real symmetric matrices whose off-diagonal sparsity pattern is the same as that of the adjacency matrix of G. In this note we present the first exact algorithm for the minimum rank…

First we construct minimal hypersurfaces $M\subset\mathbf{R}^{n+1}$ in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone $C\times \mathbf{R}$, for any strictly minimizing strictly stable cone $C$ in…

A symmetric tensor is completely positive (CP) if it is a sum of tensor powers of nonnegative vectors. This paper characterizes completely positive binary tensors. We show that a binary tensor is completely positive if and only if it…

It is shown that each positive map between matrix algebras is the sum of a maximal decomposable map and an atomic map which is both optimal and co-optimal. The result is analyzed in detail for the positive projection onto a spin factor.

Let q be a power of a prime integer p, and let X be a Hermitian variety of degree q+1 in the n-dimensional projective space. We count the number of rational normal curves that are tangent to X at distinct q+1 points with intersection…