Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.54 (2022)

Let (X,L) be a complex polarized n-fold with the structure of a geometric quadric fibration over a smooth projective surface. The Hilbert curve of (X,L) is a complex affine plane curve of degree n, containing n − 3 evenly spaced parallel lines. This paper is devoted to a detailed study of the cubic representing the residual component. Reducibility, existence of triple points, and properties of the irreducible components are analyzed in connection with the structure of (X,L).

Inspired by the work of Askey-Steinig, Szeg\"o, and Schweitzer, we provide several monotonicity theorems and inequalities for certain sine sums. Among others, we prove that for $n\geq 1$ and $x\in (0,\pi/2)$, we have $$ \frac{d}{dx} \frac{C_n(x)}{1-\cos(x)}<0 \quad\mbox{and} \quad \frac{d}{dx} \left(1-\cos(x)\right)C_n(x)>0, $$ where $$ C_n(x)=\sum_{k=1}^n\frac{\sin((2k-1)x)}{2k-1} $$ denotes Carslaw's sine polynomial. Another result states that the inequality $$ \sum_{k=1}^n (n-k+a)(n-k+b) k \sin(kx)>0 \quad (a,b\in \mathbb{R}) $$ holds for all $n\geq 1$ and $x\in (0,\pi)$ if and only if $a=b=1$. Many corollaries and applications of these results are given. Among them, we present a two-parameter class of absolutely monotonic rational functions.

Elina Robeva discovered quadratic equations satisfied by orthogonally decomposable (“odeco”) tensors. Boralevi-Draisma-Emil Horobeț-Robeva then proved that, over the real numbers, these equations characterise odeco tensors. This raises the question to what extent they also characterise the Zariski-closure of the set of odeco tensors over the complex numbers. In the current paper we restrict ourselves to symmetric tensors of order three, i.e., of format n×n×n. By providing an explicit counterexample to one of Robeva’s conjectures, we show that for n ≥ 12, these equations do not suffice. Furthermore, in the open subset where the linear span of the slices of the tensor contains an invertible matrix, we show that Robeva’s equations cut out the limits of odeco tensors for dimension n ≤ 13, and not for n ≥ 14. To this end, we show that Robeva’s equations essentially capture the Gorenstein locus in the Hilbert scheme of n points and we use work by Casnati-Jelisiejew- Notari on the (ir)reducibility of this locus.

We give a geometric description of singular pencils of quadrics of constant rank, relating them to the splitting type of some naturally associated vector bundles on P1. Then we study their orbits in the Grassmannian of lines, under the natural action of the general linear group.