Generic Newton polygons for curves of given p-rank

We survey results and open questions about the $p$-ranks and Newton polygons of Jacobians of curves in positive characteristic $p$. We prove some geometric results about the $p$-rank stratification of the moduli space of (hyperelliptic) curves. For example, if $0\le f\le g-1$, we prove that every component of the $p$-rank $f+1$ stratum of ${\mathcal{M}}_g$ contains a component of the $p$-rank $f$ stratum in its closure. We prove that the $p$-rank $f$ stratum of ${\bar{\mathcal{M}}}_g$ is connected. For all primes $p$ and all $g\ge 4$, we demonstrate the existence of a Jacobian of a smooth curve, defined over ${\bar{\mathbb{F}}}_p$, whose Newton polygon has slopes $\{0,\frac{1}{4},\frac{3}{4},1\}$. We include partial results about the generic Newton polygons of curves of given genus $g$ and $p$-rank $f$.