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**OpenURL**

##### 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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.