Rational curves on complete intersections in positive characteristic

TL;DR

This paper investigates the presence of rational curves on complete intersections in positive characteristic, demonstrating that general Calabi-Yau and general type varieties are not uniruled, with specific codimension bounds for containing rational curves.

Contribution

It proves that general Calabi-Yau and general type complete intersections in positive characteristic are not uniruled, extending known characteristic 0 results, and establishes codimension bounds for containing rational curves.

Findings

General Calabi-Yau and general type complete intersections are not uniruled in positive characteristic.

The codimension of the space of complete intersections containing a rational curve is at least _i - 2n + 2.

Counterexamples to uniruledness in positive characteristic are addressed for general cases.

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 \emph{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 $(d_1, \cdots, d_k)$ containing a rational curve has codimension at least $\sum_{i=1}^k d_i - 2n + 2$ in the moduli space of all complete intersections of given multidegree and dimension.