On the p-rank of singular curves and their smooth models

In this paper, we are concerned with the computation of the $p$-rank and $a$-number of singular curves and their smooth model. We consider a pair $X,X^{\mathrm{\prime }}$ of proper curves over an algebraically closed field $k$ of characteristic $p$, where $X^{\mathrm{\prime }}$ is a singular curve which lies on a smooth projective variety, particularly on smooth projective surfaces $S$ (with $p_g(S)=0=q(S)$) and $X$ is the smooth model of $X^{\mathrm{\prime }}$. We determine the $p$-rank of $X$ by using the exact sequence of group schemes relating the Jacobians $J_X$ and $J_{X^{\mathrm{\prime }}}$. As an application, we determine a relation about the fundamental invariants $p$-rank and $a$-number of a family of singular curves and their smooth models. Moreover, we calculate $a$-number and find lower bound for $p$-rank of a family of smooth curves.