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Abstract Heckel

Singularities of Fano varieties of lines on cubic hypersurfaces and degenerations of their Picard schemes

The subvariety $F(X)$ of the Grassmannian parameterizing the lines on a cubic hypersurface $X$ is called the Fano variety of lines on the cubic. In their 1973 paper, Clemens and Griffiths proved that the intermediate Jacobian of a smooth cubic threefold is isomorphic to the Albanese variety of the Fano variety of lines. In 2009, van der Geer and Kouvidakis studied the Picard scheme of $F(X)$ when $X$ admits at most one singular point of type $A_1$. The aim of this talk is to describe the Picard scheme of $F(X)$ when $X$ is a cubic threefold admitting isolated ADE-singularities.

I will at first give some results on $F(X)$ for a singular cubic hypersurface $X$ and, in particular, give a full description of the singularities of $F(X)$ in dependence on the singularities of $X$. Afterwards, I will review the explicit computation of degenerations of Picard schemes of curves by computing central fibres of semistable reductions with a method described by Harris and Morrison. I will show how this method can be generalized to compute degenerations of the Picard scheme of $F(X)$ at least in the simplest case. If time permits, I will discuss the general case and partial results on the way to make degenerations of the Picard scheme of $F(X)$ computable for arbitrary isolated ADE-singularities on the cubic threefold $X$.