On the period map for prime Fano threefolds of degree 10

By O. Debarre, A. Iliev, Laurent Manivel. 2008

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We prove that the deformations of a smooth complex Fano threefold X with Picard number 1, index 1, and degree 10, are unobstructed. The differential of the period map has two-dimensional kernel. We construct two two-dimensional components of the fiber of the period map through X: one is isomorphic to the variety of conics in X, modulo an involution, another is birationally isomorphic to a moduli space of semistable rank-2 torsion-free sheaves on X, modulo an involution. The threefolds corresponding to points of these components are obtained from X via conic and line (birational) transformations.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eef18a1182aa187ff1112dcb5b6d6f26cdfa284a

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Gushel--Mukai varieties: classification and birationalities (2015, background): “phic (Theorem 4.20). The birational isomorphisms relating periodpartners are generalizations ofconic transformations and those relating dual varieties are generalizations of line transformations (see [DIM]). In relation with birationalities for dual varieties, we deﬁne another interesting geometric ”
The integral Hodge conjecture for two-dimensional Calabi–Yau categories (2020, background): “Recently, GM varieties have attracted attention because of the rich structure of their birational geometry, Hodge theory, and derived categories [22, 42, 23, 25, 26, 27, 57, 76].”
Special prime Fano fourfolds of degree 10 and index 2 (2013, background): “NO FOURFOLDS 7 As in the ﬁrst case above, the kernel K is therefore isomorphic to H1(PW,Ω3 PW ⊗OPW(2)). Using (6) again, we obtain an exact sequence H1(W,Ω3 W ⊗ Sym 2 E) → K → H1(W,Ω2 W ⊗ det(E)). By [DIM1], Proposition 5.3, we have H1(W,Ω3 W ⊗ Sym 2 E) = H1(W,Ω3 W(2)) ≃ H3(W,Ω1 W(−2))∨, H1(W,Ω2 W ⊗”