Irreducibility of \(\overline{M}_{0,n}(G/P,\beta)\).

*(English)*Zbl 0934.14020From the introduction: Let \(G\) be a complex connected linear algebraic group, \(P\) be a parabolic subgroup of \(G\) and \(\beta\in A_1(G/P)\) be a 1-cycle class in the Chow group of \(G/P\). An \(n\)-pointed genus 0 stable map into \(G/P\) representing the class \(\beta\), consists of data \((\mu:C\to X;\;p_1, \dots,p_n)\), where \(C\) is a connected, at most nodal, complex projective curve of arithmetic genus 0, and \(\mu\) is a complex morphism such that \(\mu_* [C]= \beta\) in \(A_1(G/P)\). In addition \(p_i\), \(i=1,\dots,n\), denote \(n\) nonsingular marked points on \(C\) such that every component of \(C\), which by \(\mu\) maps to a point, has at least 3 points which are either nodal or among the marked points (this we will refer to as every component of \(C\) being stable). The set of \(n\)-pointed genus 0 stable maps into \(C/P\) representing the class \(\beta\) is parametrized by a coarse moduli space \(\overline M_{0,n}(G/P, \beta)\). In general it is known that \(\overline M_{0,n}(G/P,\beta)\) is a normal complex projective scheme with finite quotient singularities. In this paper we will prove that \(\overline M_{0,n}(G/P,\beta)\) is irreducible. It should also be noted that we in addition will prove that the boundary divisors in \(\overline M_{0,n}(G/P,\beta)\), usually denoted by \(D(A,B,\beta_1,\beta_2)\) \((\beta=\beta_1 +\beta_2\), \(A\cup B\) a partition of \(\{1,\dots,n\})\), are irreducible.

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14M17 | Homogeneous spaces and generalizations |

14D22 | Fine and coarse moduli spaces |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

##### Keywords:

connected projective curve; irreducibility of \(n\)-pointed genus 0 stable maps; linear algebraic group; coarse moduli space; quotient singularities##### References:

[1] | Demazure M., Desingularisation 7 pp 53– (1974) |

[2] | DOI: 10.2307/1971073 · Zbl 0309.14041 |

[3] | DOI: 10.1007/BF02568384 · Zbl 0735.14001 |

[4] | Knudsen F., Math. Scand. 52 pp 161– (1983) · Zbl 0544.14020 |

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