Surfaces on the Severi line in positive characteristic

Let \mathbf {k} be an algebraically closed field, a minimal surface X over \mathbf {k} of maximal Albanese dimension is called on the Severi line if the ‘Severi equality’: K^2_X=4\chi (\mathcal {O}_X) holds. We prove that X is on the Severi line if and only if its canonical model X_{\mathrm {can}} admits a flat double cover over an Abelian surface.