Nontriviality of the Abel-Jacobi mapping for varieties covered by rational curves (Q1913309)

Let \(X\) be a smooth complex projective variety of dimension \(n\), and let \(F\) be a smooth projective variety parametrizing a family of proper subvarieties of dimension \(d\) on \(X\). Let \(E= \{(C,x)\in F\times X: x\in C\}\), and let \(p:E \to F\) and \(q:E \to X\) be the natural projections. The cohomological Abel-Jacobi mapping is the morphism of Hodge structures of type \((-d,-d)\) defined by the composition \(H^*(X,\mathbb{C}) @>q^*>> H^*(E,\mathbb{C}) @>p_*>> H^{*-2d} (F,\mathbb{C})\). If \(D\) is an ample divisor on \(X\) then we denote by \(H^*(X,\mathbb{C})^\circ\) the primitive cohomology of \(X\) with respect to the divisor, Main result: Theorem 1.1. Let \(X\) be a proper smooth variety of dimension \(n\) that is covered by a family of rational curves, and let \(D\) be an ample divisor on \(X\). Then there exist families of rational curves on \(X\) parametrized by a smooth, possibly disconnected, nonequidimensional projective variety \(F\) of dimension \(\leq n-1\), as well as a diagram \[ \begin{aligned} & E @>\varphi>> F\times\mathbb{P}^1 @>\pi>> F \\ & q\downarrow \\ & X\end{aligned} \] such that \(E\to F\) is flat with generic fiber a rational curve, \(\varphi\) is birational, and the cohomological Abel-Jacobi mapping \((\pi\circ \varphi)_* q^*: H^n(X,\mathbb{C})^\circ\to H^{n-2} (F,\mathbb{C})\) is injective. Furthermore, each component of \(F\) parametrizes either the original covering family of rational curves or degenerations of these curves.