i will report on an ongoing joint project with david helm and yichao tian. let $p$ be a prime unramified in a totally real field $f$. the goren-oort strata are defined by the vanishing locus of the partial hasse invariants; they furnish an analog of the stratification of modular curves mod $p$ by the ordinary locus and the supersingular locus. we give an explicit global description of the goren-oort stratification of the special fiber of the hilbert modular variety for $f$. an interesting application of this result is that, when $p$ is inert of even degree, certain generalizations of the strata considered by goren-oort contribute non-trivially as tate cycles to the cohomology of the special fiber of the hilbert modular varieties. under some mild conditions, they generate all the tate cycles.