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I read a paper "Notes On Stable Maps And Quantum Cohomology, W.Fulton and R.Pandharipande". And I think that $\overline{M_{0,n}}(\mathbb{P}^n,d)$ is irreducible. But I cannot find an exact statement from any references.

It will be very grateful to tell me about reference about this, or whether some restrictions to $n,d$ is needed for the irreduciblility.

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    $\begingroup$ This is implicit in Kontsevich's original paper on "Enumeration of Rational Curves via Torus Actions" (in my opinion). It is explicit in an article by Bumsig Kim and Rahul Pandharipande, "The Connectedness of the Moduli Space of Maps to Homogeneous Spaces". It also follows from Kuznetsov's paper, "Laumon's Resolution ..." $\endgroup$
    – Jason Starr
    Commented Nov 27, 2015 at 5:11
  • $\begingroup$ Thank you for valuable comment. I'll read that papers carefully. $\endgroup$
    – free-object
    Commented Nov 27, 2015 at 10:20
  • $\begingroup$ I would like to say: Yi Zhu used and extended the results of Kim-Pandharipande in a serious way to prove rational simple connectedness (not only connectedness of $\overline{M}_{0,n}(X,\beta)$, but also simple connectedness) for all projective homogeneous spaces $X$. This had been proved for projective homogeneous spaces of Picard number $1$ by Xuhua He, but a full type-free proof of Serre's Conjecture II (the current proofs are very type-dependent) will require the case of arbitrary $X$ (and probably also wonderful compactifications as well). $\endgroup$
    – Jason Starr
    Commented Nov 27, 2015 at 12:46

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