A note on algebraic curves in characteristic 2

Abstract

Let X be a curve of genus g ≥ 3 over an algebraically closed field of characteristic 2. Let rx be the rank of the Cartier operator acting on thespace of regular differentials. We show that if rx ≤, (2g-4)/3 then g is odd and X is a double cover of a curve of genus at most rx-(g-l)/2. We also study the rank of the Cartier operator on double covers Let X be a non-singular, irreducible complete algebraic curve defined over an algebraicallyclosed field k of characteristic 2. The Cartier operator Cacts on differential forms of X by (See [3] for the proprerties of the Cartier operator on curves)C is a 1/2-linear map and it is well known that Cacts on the space of regular differentials of X. Let rx be the rank of C on this space. We shall prove the following result.