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... at least in the case where the target is a rationally connected variety.

This question is a follow-up to question Constructing embedded families of curves with general moduli and Jason Starr's suggestion in the comments there.

Let $X$ be a smooth projective variety of dimension $\geq 3$ that is separably rationally connected. Fix $g \geq 2$, $r \geq 0$ and sufficiently positive curve class $\beta$. Consider the morphism $$ \mathcal{F} := (ev,\Phi): \mathcal{M}_{g,r}(X,\beta) \to X^r \times \mathcal{M}_{g,r} $$

What can be said about the topological properties of the fiber of $\mathcal{F}$? (for a general reference to topological properties of fibers of morphism, see : Structure of fundamental groups arising from smooth projective morphisms)

That is,

  1. What conditions do I need to impose to say something about the homotopy type of the general fiber? In particular, what restrictions can I impose to deduce that it is connected and simply connected outside a complex codimension 1 subset of $\mathcal{M}_{g,r} \times X^r$? The motivation for this question is to understand if we can lift classes in $\pi_1^{orb}(\mathcal{M}_{g,r})$ to $\pi_1(\mathcal{M}_{g,r}(X,\beta))$ using the standard fibration long exact sequence argument.
  2. I know the study of the rational connectedness of the fibers (which I guess is kind of an algebraic-geometry analogue of connectedness) of these map plays some kind of a rule in the study of rational connectedness of the target variety. Can somehow provide a reference for a good paper that summerizes what is currently known?
  3. Is there a known paper that studies the lift of the orbifold fundemental group of $\mathcal{M}_{g,r}$ in this context? That is, a lift of the MCG to the fundamental group of the Kontsevich moduli space of stable maps?
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    $\begingroup$ For $g$ equal to $0$ and for $r=1$, there is a conjecture of Graeme Segal, refined by Cohen-Jones-Segal, that as $\langle c_1(T_X),\beta \rangle \to \infty$, the homotopy types of the fibers of $\text{ev}:\mathcal{M}_{0,1}(X,\beta) \to X$ converge (in an appropriate sense) to the homotopy type of the space $\text{Hom}_{\text{ptd}}((\mathbf{S}^2,0),(X^\text{an},{x_0})$, the double loop space of $X$. Segal proved this for $X=\mathbb{P}^n$, and there were many cases proved after that, cf. "Stability for holomorphic spheres and Morse theory" , Cohen, Jones, Segal, and the references therein. $\endgroup$ – Jason Starr Mar 21 '17 at 13:57
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    $\begingroup$ There are two sources of a nontrivial fundamental group of the space of holomorphic spheres. First of all, nontrivial elements of $\pi_3(X^{\text{an}})$ give nontrivial elements of $\pi_1$ of the double loop space. For $X$ Fano, most such elements arise from surjective morphisms $\pi:X\to \mathbb{P}^1$. Every such morphism factors through a morphism with connected fibers. By the Rationally Connected Fibration Theorem, there exists a section $s:\mathbb{P}^1\to X$. Thus, $\pi_3(X^{\text{an}})\to \pi_3(\mathbb{CP}^1)$ is surjective, so the Hopf class in $\pi_3(\mathbb{CP}^1)$ lifts . . . $\endgroup$ – Jason Starr Mar 21 '17 at 14:09
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    $\begingroup$ . . . The corresponding contribution to the fundamental group of $\mathcal{M}_{0,1}(X,\beta)$ works in the same way: stabilization with respect to $\pi$ defines a morphism $\mathcal{M}_{0,1}(X,\beta) \to \mathcal{M}_{0,1}(\mathbb{P}^1,\pi_*\beta)$, and the section $s$ defines a section of this (for appropriate $\beta$). The fundamental group of $\mathcal{M}_{0,1}(\mathbb{P}^1,e)$ can be understood in terms of monodromy about the boundary components in the simply connected space $\overline{\mathcal{M}}_{0,1}(\mathbb{P}^1,e)$, but it is basically the Hopf class again . . . $\endgroup$ – Jason Starr Mar 21 '17 at 14:13
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    $\begingroup$ . . . The second source for the fundamental group comes from $H^{2,1}(X)$, i.e., a nontrivial Griffiths intermediate Jacobian $J_1(X)$. In this case, there is an Abel map $\alpha:\mathcal{M}_{0,1}(X,\beta)\to J_1(X)$, and this is dominant for sufficiently positive $\beta$. So the nontrivial fundamental group of the complex torus $J_1(X)$ implies that also $\mathcal{M}_{0,1}(X,\beta)$ is nontrivial. Segal's conjecture implies, more or less, these are the only contributions to the fundamental group as $\beta \to \infty$, cf. the discussion of $\pi_3(X)$ in the book by Griffiths-Morgan. $\endgroup$ – Jason Starr Mar 21 '17 at 14:18
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    $\begingroup$ For $g\geq 1$, the fibers of $\mathcal{M}_{g,r}(X,\beta)\to X^r \times \mathcal{M}_{g,r}$ will not be simply connected. For every invertible sheaf $\mathcal{L}$ on $X$, for every $r$-pointed genus $g$ curve $(C,p_1,\dots,p_r)$, for every morphism $u:C\to X$, the class of $u^*\mathcal{L}$ defines an element in the Jacobian of $C$. So the fibers of the forgetful map have nontrivial Abel maps. You could factor the forgetful map through a universal Abel map to a relative Picard over $\mathcal{M}_{g,r}$ and try to prove that the fibers of that Abel map are simply connected. $\endgroup$ – Jason Starr Mar 21 '17 at 15:09

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