Effective Divisors on Kontsevich Moduli Spaces

We study the cone of effective divisors on Kontsevich's moduli space of genus 0 stable maps to projective space in the case where map is equipped with a marked point on the domain curve, extending previous work of Coskun, Harris, and Starr. This moduli space is equipped with a canonical evaluation morphism to projective space, the fibers of which are the simplest examples of Gromov-Witten varieties. We show that the Picard group and the effective cone of this variety follow from the corresponding computations on the moduli space in the simplest way for which one could hope.