On nodal prime Fano threefolds of degree 10

Abstract

We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and degree 10. We show that these threefolds are birationally isomorphic to Verra threefolds, i. e., hypersurfaces of bidegree (2,2) in P2 × P2. Using Verra's results on the period map for these threefolds and on the Prym map for double étale covers of plane sextic curves, we prove that the fiber of the period map for our nodal threefolds is the union of two disjoint surfaces, for which we give several descriptions. This result is the analog in the nodal case of a result of Debarre O, Iliev A, Manivel L (arXiv: 0812.3670) in the smooth case. © 2011 Science China Press and Springer-Verlag Berlin Heidelberg.