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For curves of genus bigger than one we prove that Buium's first arithmeticjet spaces (p-jet spaces) admit the structure of a torsor under some linebundle.  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 Frobeniustangent bundle so that the first $p$-jet space (and hence higher order lifts ofthe Frobenius) form torsor a under this bundle.  The Cech cohomology classes associated to this torsor structure, which wecall the Deligne-Illusie class, has strong analogies with the classicalKodaira-Spencer class from deformation theory.

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