The Abel-Jacobi mapping for the quartic threefold
Let \(X\) be a generic quartic 3-fold and let \(F(X)\) be the Fano surface of conics lying on it. This paper is devoted to prove that the Abel-Jacobi map \(\Phi_{F(X)}: A(F(X))\to J(X)\), where \(A(F(X))\) is the Albanese variety of \(F(X)\) and \(J(X)\) is the intermediate Jacobian of \(X\), is an isomorphism. To this aim the author studies in detail the surface \(F(X)\), showing in particular that, if \(\{X_ t\}_{t\in\mathbb P^ 1}\) is a generic pencil of quartic 3-folds and \(X_ t\) is smooth for \(t\in\mathbb P^ 1-\{t_ 1,\ldots,t_ n\}\), there exist \(t_{n+1},\ldots,t_ m\in\mathbb P^ 1\) such that: (i) \(F(X_ t)\) is smooth and irreducible for \(t\in\mathbb P^ 1-\{t_ 1,\ldots,t_ m\}\); (ii) \(F(X_{t_{n+j}})\), \(1\leq j\leq m-n\), has only isolated singularities; (iii) \(F(X_{t_ i})\), \(1\leq i\leq n\), has as only singularities an irreducible smooth curve \(\Gamma_ i\) of double points of rank 2, and locally around \(\Gamma_ i\), \(F(X_{t_ i})\) is analytically reducible to two isomorphic smooth components. Then the required isomorphism is deduced from these facts by means of topological arguments, involving monodromy and vanishing cycles; this part of the proof is based on results of \textit{H. Clemens} [Ann. Math. (2) 117, 71--76 (1983; Zbl 0512.14004)]. It is known that the same result holds when \(X\) is a generic cubic 3-fold and \(F(X)\) is the family of lines lying on it [\textit{C. H. Clemens} and \textit{P. A. Griffiths}, Ann. Math. (2) 95, 281--356 (1972; Zbl 0214.48302)]. For a survey of the known results on this subject see also: \textit{H. Clemens}, ``Some results about Abel-Jacobi mappings'' in: Topics in transcendental algebraic geometry, Ann. Math. Stud. 106, 289--304 (1984; Zbl 0575.14007).
scientific article; zbMATH DE number 3910543
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