Transversality Properties on the Moduli Space of Genus 0 Stable Maps to a Smooth Rational Projective Surface and Their Real Enumerative Implications

We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a rational projective surface. This characterization is based on the singularity analysis of the product of evaluation maps and deformation properties of the pointed stable maps. If a target space is equipped with a real structure, i.e, anti-holomorphic involution, then this transversality result has a real enumerative implication. One of the natural consequence of this result is Welschinger’s invariant in an algebraic geometry category, which is about the global minimum bound of the number of real stable maps passing through certain real points in a general position on the target space.