  Hartshorne conjectured that a smooth, codimension c subvariety of
n-dimensional projective space must be a complete intersection, whenever c is
less than n/3. We prove this in the special case when n is much larger than the
degree of the subvariety. Similar results were known in characteristic zero due
to Hartshorne, Barth-Van de Ven, and others. Our proof is field independent and
employs quite different methods from those previous results, as we connect
Hartshorne's Conjecture with the circle of ideas initiated by Ananyan and
Hochster in their proof of Stillman's Conjecture.
