On nodal prime Fano threefolds of degree 10

TL;DR

This paper investigates the geometry and period map of nodal prime Fano threefolds of degree 10, establishing their birational relation to Verra solids and analyzing the structure of their period map fibers.

Contribution

It demonstrates that nodal prime Fano threefolds of degree 10 are birationally equivalent to Verra solids and characterizes the fiber of their period map as a union of two surfaces.

Findings

Nodal prime Fano threefolds of degree 10 are birational to Verra solids.

The period map fiber for these threefolds is a union of two surfaces.

The results extend known smooth case properties to the nodal case.

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 solids (hypersurfaces of bidegree $(2,2)$ in $ \P^2\times \P^2$). Using Verra's results on the period map for these solids and on the Prym map for double \'etale covers of plane sextic curves, we prove that the fiber of the period map for our nodal threefolds is birationally the union of two surfaces, for which we give several descriptions. This result is the analog in the nodal case of a result obtained in arXiv:0812.3670 for the smooth case.