19 research outputs found
The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14
The Abel-Jacobi maps of the families of elliptic quintics and rational
quartics lying on a smooth cubic threefold are studied. It is proved that their
generic fiber is the 5-dimensional projective space for quintics, and a smooth
3-dimensional variety birational to the cubic itself for quartics. The paper is
a continuation of the recent work of Markushevich-Tikhomirov, who showed that
the first Abel-Jacobi map factors through the moduli component of stable rank 2
vector bundles on the cubic threefold with Chern numbers
obtained by Serre's construction from elliptic quintics, and that the
factorizing map from the moduli space to the intermediate Jacobian is \'etale.
The above result implies that the degree of the \'etale map is 1, hence the
moduli component of vector bundles is birational to the intermediate Jacobian.
As an applicaton, it is shown that the generic fiber of the period map of Fano
varieties of degree 14 is birational to the intermediate Jacobian of the
associated cubic threefold.Comment: Latex, 28 page
Holomorphic symplectic geometry: a problem list
A list of open problems on holomorphic symplectic, contact and Poisson
manifolds