Narrow Results

For an algebraic curve ${𝒳}$ defined over an algebraically closed field of characteristic $p>0$, $a$-number $a({𝒳})$ is the dimension of the space of exact holomorphic differentials on ${𝒳}$. We computed the $a$-number for a family of certain Picard curves using the action of the Cartier operator on $H^0({𝒳},{\mathrm{\Omega }}^1)$.