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 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

Quartic del Pezzo surfaces over function fields of curves

“…Let X = D ∩ Q where Q has bidegree (0 2), i.e., Q is the pullback of a quadric from P 6 . The image of P(V ∨ ) in P 6 is a quadric hypersurface.…”

“…Characterize sets of points ( 1 6 ) ∈ (P 1 ) 6 such that there exists some degree-two morphism ψ : P 1 → P 1 with…”

“…This image is unirational as it is dominated by P 5 × P 6  the choice of a plane conic and a plane cubic with a node at a prescribed point.…”

“…height −2: the canonical section σ (P 1 ); height −1: the sections arising from exceptional fibers over points of C , which depend on one parameter; height 0: the sections arising from secant lines to C ⊂ P 3 , which are parametrized by Sym 2 (C ); height 1: the sections arising from conics five-secant to C , which are parametrized by Sym 3 (C ); height 2: the sections arising from twisted cubics meeting C in eight points, which are parametrized by Sym 4 (C ); height 3: the sections arising from eleven-secant quartic rational curves, which are parametrized by Sym 5 (C ); height 4: the sections arising from quintic rational curves 14-secant to C , which are parametrized by Sym 6 (C ); height 5: the sections arising from sextic rational curves 17-secant to C , which are parametrized by Sym 7 (C ).…”

“…Thus R corresponds to a section of π; these are parametrized by elements of Sym 6 (C ), which may be interpreted as the theta divisor of J(C ).…”