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After a brief review of the kinematics of deep inelastic scattering (DIS)within the so-called $\Sigma$ method, we derive the necessary formulae for thetreatment of QED radiative corrections to DIS originating from hard photonradiation. The results are applied to a calculation of the corrections to DISwith a tagged photon with next-to-leading logarithmic accuracy under HERAconditions. It turns out that the next-to-leading order corrections are quiteimportant for the $\Sigma$ method. We also discuss the dependence of thecorrections on the longitudinal structure function of the proton, $F_L$, in theregion of low $Q^2$ and moderate $x$.Comment: 28 pages, LaTeX2e, 7 figs. Labels of fig.6 fixed (correct in journal

Deep inelastic scattering with a tagged photon: QED corrections for the