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Real aspects of the moduli space of genus zero stable maps

We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space. | Turk J Math 31 (2007) , 303 – 317. ¨ ITAK ˙ c TUB Real Aspects of the Moduli Space of Genus Zero Stable Maps Seongchun Kwon Abstract We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space. Key Words: Moduli space of genus zero stable maps, real variety, real structure. 1. Introduction We call a projective variety V as a real projective variety if V has an anti-holomorphic involution τ on the set of complex points V (C). By a real structure on V , we mean an anti-holomorphic involution τ . The real part of (V, τ ) is the locus which is fixed by τ . In the following paragraph, readers can find the definitions of the moduli space of stable maps and various maps defined on it in [3]. Let’s assume that X is a convex real projective variety. We show the following: • The moduli space M k (X, β) of k-pointed genus 0 stable maps is a real projective variety. • Let M k be the Deligne-Mumford moduli space of k-pointed genus 0 curves. The forgetful maps Fn : M n (X, β) → M n−1 (X, β), F : M k (X, β) → M k are real 2000 AMS Mathematics Subject Classification: 14N10, 14N35, 14P99 303 KWON morphisms(i.e., morphisms which commute with the anti-holomorphic involution on the domain and that on the target), where n ≥ 1, k ≥ 3. • Let evi : M k (X, β) → X be the i-th evaluation map. Then, evi is a real map. • Let CPn have the real structure from the complex conjugation involution. Let X be a real projective variety such that the imbedding i which decides the real structure on X has a non-empty intersection with RPn ⊂ CPn . Let Mk (X, β), k ≥ 3, be the moduli space of k-pointed genus 0 stable maps with a smooth domain curve. Then, each point in the real part Mk (X, β)re of the moduli space represents a real stable map having marked points on the real part of the domain curve. This