We prove that every complete family of linearly non-degenerate rational curves of degree $e > 2$ in $\mathbb{P}^{n}$ has at most $n-1$ moduli. For $e = 2$ we prove that such a family has at most $n$ moduli. The general method involves exhibiting a map from the base of a family $X$ to the Grassmannian of $e$-planes in $\mathbb{P}^{n}$ and analyzing the resulting map on cohomology.

A new tautological relation of $\overline{\mathcal{M}}_{3,1}$ in codimension 3 is derived and proved, using an invariance constraint from our previous work.

In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, $\Kgnb{0,0}(\PP^r,d)$, stabilize when $r \geq d$. We give a complete characterization of the effective divisors on $\Kgnb{0,0}(\PP^d,d)$. They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

The purpose of this note is to give a simple combinatorial construction of the map from the canonically compactified moduli spaces of punctured complex projective lines to the moduli spaces $\P_r$ of polygons with fixed side lengths in the Euclidean space $\E^3$. The advantage of this construction is that one can obtain a complete set of linear relations among the cycles that generate homology of $\P_r$. We also classify moduli spaces of pentagons.