  We study intersection theory for differential algebraic varieties.
Particularly, we study families of differential hypersurface sections of
arbitrary affine differential algebraic varieties over a differential field. We
prove the differential analogue of Bertini's theorem, namely that for an
arbitrary geometrically irreducible differential algebraic variety which is not
an algebraic curve, generic hypersurface sections are geometrically irreducible
and codimension one. Surprisingly, we prove a stronger result in the case that
the order of the differential hypersurface is at least one; namely that the
generic differential hypersurface sections of an irreducible differential
algebraic variety are irreducible and codimension one. We also calculate the
Kolchin polynomials of the intersections and prove several other results
regarding intersections of differential algebraic varieties.
