Deligne-Illusie Classes I: Lifted Torsors of Lifts of the Frobenius for Curves

For curves of genus bigger than one we prove that Buium's first arithmetic jet spaces (p-jet spaces) admit the structure of a torsor under some line bundle. This result lifts a known constructions in characteristic p where the first $p$-jet space modulo p is a sheaf under the Frobenius tangent sheaf (parametrizing Frobenius linear derivations). In particular we show there is a natural family of lifts of the Frobenius tangent bundle so that the first $p$-jet space (and hence higher order lifts of the Frobenius) form torsor a under this bundle. The Cech cohomology classes associated to this torsor structure, which we call the Deligne-Illusie class, has strong analogies with the classical Kodaira-Spencer class from deformation theory.