Geometry

The problem of finding the maximum symmetric rank in dimension three—which, to our knowledge, is known only for degrees up to five—aims at a deeper understanding of tensor rank, especially in view of its computational complexity applications. We test a conjectural answer by pursuing a strategy that has led to the best known upper bound to date. As a first outcome, our main result reveals a connection between Waring decompositions and certain spectral data. We then highlight some examples of quartics that exhibit a special behavior. Finally, our findings indicate that a different conjectural answer, presented here for the first time, may be more plausible.

A polygon is equable if its area is equal to its perimeter. A pair of polygons is an amicable pair if the area of the first is equal to the perimeter of the second, and vice versa. A polygon is a lattice polygon if its vertices lie on the integer lattice. We show that amicable lattice rhombuses are actually equable.

Let X be a smooth minimal complex surface of general type subject to the inequality The old problem of Severi states that the image of the Albanese map of such a surface is at most a curve. The problem was completely settled by Rita Pardini using the works of Xiao on fibered surfaces. The paper suggests a different approach to Severi type inequalities through the study of the cohomology group $H^1\left({\Theta }_X(-K_X)\right)$. For X irregular, with no irrational pencil, it is shown that the cohomology classes of the group carry geometric data: where ${\mathcal{F}}_{\xi }$ is a rank 2 bundle, is a modification of ${\Omega }_X$ along an effective, nonzero divisor $E_{\xi }$ inducing the isomorphism on the level of global sections: $A_{\xi }$ is at most 0-dimensional subscheme of $E_{\xi }$. The paper establishes many geometrical properties of $E_{\xi }$ and $A_{\xi }$. As a consequence, for an irregular surface X without irrational pencil, sufficient conditions are given for a Severi type inequality where and $q=h^0\left({\Omega }_X\right)$ are the Chern numbers and the irregularity of X, respectively. The dependence of the geometric data on $\left[\xi \right]$ is explored: it is shown that there is a finite collection of projective subspaces $\left\{\mathbb{P}\left(W_E\right)\right\}$ of , labeled by effective nonzero divisors E; the data are constant on the complement of the collection and undergoes a change on each $\mathbb{P}\left(W_E\right)$. One attaches a quiver to this collection which is an interesting object to study.