The present work exploits simplifications arising in weakly exothermic detonations when the postshock conditions are supersonic to investigate the structure of wedge-induced oblique detonations. These simplifications enable the linearized Euler equations (employed here in characteristic form) to be efficiently solved numerically, subject to the linearized Rankine–Hugoniot jump conditions across the leading oblique shock. A first set of computations employs one-step first-order Arrhenius chemistry appropriate for describing detonations when the postshock chemistry exhibits a thermal-explosion character. In that case, the relevant chemical-kinetic parameter of order unity $\beta$ is the product of the heat release and the activation energy divided by the square of the postshock thermal enthalpy. The transition from the shock to the detonation wave is continuous at small $\beta$, begins to develop spatially decaying oscillations as $\beta$ increases, and develops a singularity at the shock at a critical value of $\beta$; above which, the transition must become discontinuous and involve a triple point. Parametric results are presented in a plane of the wedge angle and the incident-flow Mach number: the two important controlling parameters. The triple point is found to develop when the incident-flow Mach number falls below a critical value that exhibits a U-shaped dependence on the wedge angle, becoming large at both high and low wedge angles and reflecting large differences between shock angles with and without heat release in those two extremes. Additional computations are performed for a three-step branched-chain scheme with the heat-release step having zero activation energy and for very fuel-lean hydrogen–air detonations with postshock temperatures above crossover. These cases, for which ignition develops as a chain-branching explosion, do not develop a singularity at the shock; although they display many of the features identified with the Arrhenius chemistry, including oscillations and appearance of a precursor point indicative of criticality. The results suggest a strong potential influence of the chemistry on the transition.