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On nodal prime Fano threefolds of degree 10. (English) Zbl 1247.14043

Start with a complex Fano \(3\)-fold \(X\) with Picard group \({\mathbb Z}[K_X]\) and degree \((-K_X)^2=10\) having a single node. If \(X\) is general enough it has two distinct conic bundle structures, described here. Both arise from taking the anticanonical embedding of \(X\) in \({\mathbb P}^7\). Then the projection from the node is birational onto a singular complete intersection of three quadrics in \({\mathbb P}^6\), which in turn is birational to a conic bundle whose discriminant is the union of a line \(\Gamma_1\) and a smooth plane sextic \(\Gamma_6\). This determines an étale double cover \(\pi: \tilde\Gamma_6\cup\Gamma'_1\cup\Gamma''_1 \to \Gamma_6\cup\Gamma_1\). Alternatively, projection from the Zariski tangent space at the node in \({\mathbb P}^7\) also gives a conic bundle, this time with discriminant a smooth plane sextic \(\Gamma^\star_6\), and hence an étale double cover \(\pi^\star:\tilde\Gamma^\star_6\to \Gamma^\star_6\).
These two structures combine to give a birational map to a degree \((2,2)\) hypersurface \(T\subset{\mathbb P}^2\times{\mathbb P}^2\). The curves \(\Gamma_6\) and \(\Gamma^\star_6\) are not isomorphic, but the Pryms of \(\pi\) and \(\pi^\star\) are both isomorphic to the intermediate Jacobian of \(T\). A theorem of A. Verra [in: The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871–1952), Torino, Italy, September 29–October 5, 2002. Torino: Università di Torino, Dipartimento di Matematica. 735–759 (2004; Zbl 1169.14310)] says that the Prym map on plane sextics has degree \(2\), so we have found both possibilities.
The authors use this and results of D. Logachev [“Fano threefolds of genus 6”, arXiv:math/0407147] (that paper was written in 1982 and posted on arxiv.org in 2004, but apparently never published) to describe the fibres of the period map, i.e.the nodal \(3\)-folds whose intermediate Jacobian agrees with the intermediate Jacobian of \(X\). The fibre consists of two distinct surfaces, described fully here.
Obviously the hope is that by studying this degenerate case one can also describe the fibres of the period map for smooth prime Fano \(3\)-folds of degree \(10\). This question and the associated geometry have already been studied by the same authors [J. Algebr. Geom. 21, No. 1, 21–59 (2012; Zbl 1250.14029)], but at present it is still not possible, because of properness issues, to pass from the special fibres to the general ones.

MSC:

14J45 Fano varieties
14C34 Torelli problem
14D22 Fine and coarse moduli spaces
14E05 Rational and birational maps
14E30 Minimal model program (Mori theory, extremal rays)
14H40 Jacobians, Prym varieties
14J10 Families, moduli, classification: algebraic theory
14J30 \(3\)-folds
14M10 Complete intersections
14M15 Grassmannians, Schubert varieties, flag manifolds
32G20 Period matrices, variation of Hodge structure; degenerations
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References:

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