On the period map for prime Fano threefolds of degree 10

Abstract

We study, after Logachev, the geometry of smooth complex Fano threefolds X X with Picard number 1 1 , index 1 1 , and degree 10 10 , and their period map to the moduli space of 10-dimensional principally polarized abelian varieties. We prove that a general such X X has no nontrival automorphisms. By a simple deformation argument and a parameter count, we show that X X is not birational to a quartic double solid, disproving a conjecture of Tyurin. Through a detailed study of the variety of conics contained in X X , a smooth projective irreducible surface of general type with globally generated cotangent bundle, we construct two smooth projective two-dimensional components of the fiber of the period map through a general X X : one is isomorphic to the variety of conics in X X , modulo an involution, another is birationally isomorphic to a moduli space of semistable rank- 2 2 torsion-free sheaves on X X , modulo an involution. The threefolds corresponding to points of these components are obtained from X X via conic and line (birational) transformations. The general fiber of the period map is the disjoint union of an even number of smooth projective surfaces of this type.

On nodal prime Fano threefolds of degree 10

“…On the one hand, we obtain the analog in the nodal case of the reconstruction theorem of [9, Theorem 9.1]: a general nodal X can be reconstructed from the surface F g (X). On the other hand, the present description of the fiber of the period map at a nodal X fits in with the construction in [9] of two (proper smooth) surfaces in the fiber of the period map at a smooth X , one of them is isomorphic to F m (X )/ι (see [9,Theorem 6.4]). In both the smooth and nodal cases, the threefolds in the fiber of the period map are obtained one from another by explicit birational transformations called line and conic transformations (see Subsections 4.3 and 5.5).…”

“…, 10, 12}. The article is a sequel to [9], where we studied the geometry and the period map of smooth complex Fano threefolds X with Picard group Z[K X ] and degree 10. Following again Logachev [15,Subsection 5], we study here complex Fano threefolds X with Picard group Z[K X ] and degree 10 which are general with one node.…”

Gushel-Mukai varieties: Classification and birationalities

“…The birational isomorphisms relating period partners are generalizations of conic transformations, and those relating dual varieties are generalizations of line transformations [DIM12]. In relation with birationalities for dual varieties, we define another interesting geometric object associated with GM varieties: a special hypersurface of degree 4 in Gr(3, 6).…”

Quartic del Pezzo surfaces over function fields of curves

“…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 ).…”

“…Then R is the residual to R in T ∩ S. The intersection of R with C consists of seven points, which gives us our point in Sym 7 (C ).…”