M-regularity of the Fano surface

Details M-regularity of the Fano surface M-regularity of the Fano surface

2007-04-04

arxiv.org/abs/0704.0558v1

Mathematics

Algebraic Geometry

5 pages

Scientific paper

Let (A,\Theta) be a principally polarised abelian variety, and let Y be a subvariety. Pareschi and Popa conjectured that Y has minimal cohomology class if and only if the structure sheaf of Y satisfies a property that they call M-regularity. Let now X be a smooth cubic threefold. By a classical result due to Clemens and Griffiths, its intermediate Jacobian J(X) is a principally polarised abelian variety; furthermore the Fano surface of lines on X can be embedded in J(X) and has minimal cohomology class. In this short note we show that its structure sheaf is M-regular.

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