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Irreducibility of $$\overline{M}_{0,n}(G/P,\beta)$$. (English) Zbl 0934.14020
From 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
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##### 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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