Stability of logarithmic tangents

Abstract: To a hypersurface $D$ of projective $N$-space $\mathbb{P}^N$ one attaches the \emph{log tangent sheaf} $T_D$ of vector fields of $\mathbb{P}^N$ tangent to D. For some highly special hypersurfaces, such as, for instance, hyperplane arrangements associated to reflection groups and discriminants of binary forms, the sheaf $T_D$ splits into line bundles - $D$ is then called \emph{free}. On the other hand, Dolgachev--Kapranov proved that $T_D$ is stable if $D$ is a generic arrangement of at least $N+2$ hyperplanes; also Dimca proved that $T_D$ is stable if $D$ has isolated singularities with sufficiently small Tjurina number and $N=3$.

In this talk we will first show that $T_D$ is stable for a much wider class of hypersurfaces having low-dimensional singularities. In the second part of the talk we will prove that $T_D$ is stable if $D$ is the determinant of $n\times n$ matrices. If time allows, we will discuss the application from the equisingular Hilbert scheme containing $D$ to the moduli space of semistable sheaves containing $T_D$ and show that it is birational in the case of determinants.

This is a report on work in progress with S. Marchesi - soon on the arXiv.

algebraic geometry

Audience: researchers in the topic

Brazilian algebraic geometry seminar

Series comments: Subscribe the seminar mailing list, please send and email to brag-seminar-request@lists.ime.unicamp.br with "subscribe" in the subject line.

Previous talks available at the YouTube channel "Brazilian Algebraic Geometry" www.youtube.com/channel/UCM-pcdNdpWxQFgOg-illE2w