Résume | There are a natural collection of rational curves (deep) in theboundary of $\overline{M}_{g,n}$, the so called 1-strata, theirreducible components of the locus of (pointed) curves with at least3g-3 +n -1 singular points. Together with Gibney and Morrison, Iconjecture that a divisor is ample iff it has positive intersectionwith each of these curves, and moreover, in char p, that a divisor issemi-ample (ie. the linear system of some positive multiple is freeof base-points) iff it has non-negative intersection with each ofthese curves. I'll explain our main result, which is that theconjecture holds in general iff it holds for $g=0$, and strongevidence for the $g=0$ case. |