Yilong Zhang : Hilbert schemes of skew lines on cubic threefolds

Posted by Izzet Coskun , part of the Algebraic Geometry Seminar.

At

March 29, 2021, 3 p.m.

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Abstract

For a smooth cubic threefold Y, its Hilbert scheme with Hilbert polynomial 2n+2 has two irreducible components H and H'. The general member for H is a pair of skew lines and a general member for H' is a conic union an isolated point. We will show that the component H is smooth and is isomorphic to the blow-up of the 2nd symmetric product of Fano surface of lines on Y along the diagonal. This work is based on the work on Hilbert schemes of skew lines on projective spaces by Chen, Coskun and Nollet in 2011. Moreover, I'll also explain the relation of the component H to the stable moduli space considered by Altavilla-Petkovic-Rota and the compactification of locus of vanishing cycles on hyperplane sections.