# On the stability of logarithmic tangent sheaves

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Simone Marchesi

created by bazzoni on 16 Jan 2021

modified on 16 Mar 2021

17 mar 2021
-- 16:00

Geometry in Como - Online

The talk will take place on Microsoft Teams; you find the link below. Please visit https://sites.google.com/site/gbazzoni/geometry-at-insubria and fill in the form if you'd like to be added to our mailing list.

**Abstract.**

Given a hypersurface $D$ in the projective space $\mathbb{P}^N$, we can associate to it its *logarithmic tangent sheaf* $\mathcal{T}_D$, which is given by the vector fields of $\mathbb{P}^N$ that are tangent to $D$.

For particular families of hypersurfaces, such reflexive sheaf turns out to be a direct sum of line bundles and, in this case, $D$ is called *free*. This situation has been of special interest in the topic of *hyperplane arrangements*.

Going on the "opposite direction", other interesting classes of hypersurfaces give us a stable sheaf $\mathcal{T}_D$. We recall, among many, the work of Dolgachev-Kapranov, where stability is proven if $D$ is an hyperplane arrangement of at least $N+2$ hyperplanes, or the work of Dimca, where it is proven for $D\subset \mathbb{P}^3$ with isolated singularities and small Tjurina number.

In this talk, we will extend the study of stability to a wider family of hypersurfaces, relating it to the degree and dimension of the singular locus of $D$. Furthermore we will show that stability holds for the hypersurfaces defined by determinants. Finally, for this last set, we will describe the moduli map from the quotient which describes the matrices whose determinant defines $D$ and the moduli space of semistable shaves on $\mathbb{P}^N$ that contains $\mathcal{T}_D$.

This is a joint work with Daniele Faenzi.