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We construct regular components of the moduli space of stable maps from curves of genus $g$ to a product of two projective spaces. These components are generically smooth and have the expected dimension predicted by deformation theory. This result can be seen as a general position theorem for loci in $M_g$ consisting of curves carrying exceptional linear series. [ABSTRACT FROM AUTHOR]

We compute the Mori cones of the moduli spaces $\overline M_{g,n}$ of $n$ pointed stable curves of genus $g$, when $g$ and $n$ are relatively small. For instance we show that for $g<14$ every curve in $\overline M_g$ is equivalent to an effective combination of the components of the locus of curves with $3g-4$ nodes. We completely describe the cone of nef divisors for the space $ \overline M_{0,6}$, thus verifying Fulton's conjecture for this space. Using this description we obtain a classification of all the fibrations of $\overline M_{0,6}$. [ABSTRACT FROM AUTHOR]