Rational curves on complete intersections in positive characteristic

Abstract

We study properties of rational curves on complete intersections in positive characteristic. It has long been known that in characteristic 0, smooth Calabi–Yau and general type varieties are not uniruled. In positive characteristic, however, there are well-known counterexamples to this statement. We will show that nevertheless, a general Calabi–Yau or general type complete intersection in projective space is not uniruled. We will also show that the space of complete intersections of degree (d1,⋯,dk) containing a rational curve has codimension at least ∑i=1kdi−2n+2 and give similar results for hypersurfaces containing higher genus curves.