This paper presents results of a theoretical investigation of the combustion of a carbon surface in an oxidizing cross flow in the stagnation point region. A proceeding study by the authors, which has mainly been concerned with the influence of the free stream parameters on the rate of combustion, is extended to the investigation of ignition and quench phenomena leading to discontinuous behavior of the steady state solutions. With respect to experimental studies of the stagnation point combustion of carbon, which are based on different mechanisms of controlling the combustion process, two of these mechanisms are distinguished in this paper. On the one hand an externally generated heat flux to the surface may control the process, and on the other hand the surface temperature may be the controlling or independent variable. The ignition and quench behavior strongly depends upon these controlling mechanisms. In the first case up to five steady state solutions may be obtained corresponding to different heterogeneously or homogeneously ignited or quenched states. If the surface temperature is tlae controlling parameter no heterogeneous ignition or quenching occurs. Then only up to three steady state solutions are possible, which correspond to an ignited or quenched homogeneous reaction. The sensitivity of the ignition and quench behavior to variations in the free stream parameters and in the heterogeneous reaction rate data is investigated with respect to both controlling mechanisms. Emphasis is put on the influence of the Boudouard reaction rate data, which are rather uncertain.