impanga 2024/25

I will present the recent developments on free plane curves, mostly focusing on the problem of constructing algebraic surfaces having large Picard numbers and the so-called Numerical Terao's Conjecture. Time permitting, I will deliver some recent progress on Ziegler pairs of line arrangements.

The r-th cactus variety of a subvariety X in a projective space generalises secant variety of X and it is defined using linear spans of finite schemes of degree r. It's original purpose was to study the vanishing sets of catalecticant minors. We propose adding a scheme structure to the cactus variety and we define it via relative linear spans of families of finite schemes over a potentially non-reduced base. In this way we are able to study the vanishing scheme of the catalecticant minors. For X which is a sufficiently large Veronese reembedding of projective variety, we show that r-th cactus scheme and the zero scheme of appropriate catalecticant minors agree on an open and dense subset which is the complement of the (r-1)-st cactus variety/scheme. As an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties. Based on a joint work with Hanieh Keneshlou.

The existence of Kaehler-Einstein metrics on Fano 3-folds can be determined by studying lower bounds of stability thresholds. An effective way to verify such bounds is to construct flags of point-curve-surface inside the Fano 3-folds. This approach was initiated by Abban-Zhuang, and allows us to restrict the computation of bounds for stability thresholds only on flags. We employ this machinery to prove K-stability of terminal quasi-smooth Fano 3-fold hypersurfaces. This is deeply intertwined with the geometry of the hypersurfaces: in fact, birational rigidity and superrigidity play a crucial role. The superrigid case had been attacked by Kim-Okada-Won. In this talk I will discuss the K-stability of strictly rigid Fano hypersurfaces via Abban-Zhuang Theory. This is a joint work with Takuzo Okada.

A fundamental problem in the theory of tensors is establishing lower bounds for the border ranks of explicit tensors. In the talk I will present a technique for establishing lower bounds on border rank based on border apolarity and I will discuss progress concerning the border rank of the 3x3 matrix multiplication tensor. The talk is based on a joint work with Amy Huang, Austin Conner and J.M. Landsberg.