There are several layers of techniques and tools at your disposal to keep you from dying in such a situation. First, if you fall you should "self-arrest" which means turning over so you are upright (falling feet first) and facing the ground. Then you can use your ice ax in a standard maneuver with one hand on the back of the head held near your shoulder and the other hand on the end of the handle held near your belly, you then dig the head of the ax into the ice or snow and pull up on the handle so the ax bites into the snow/ice deeper and harder, until you stop. This is all one move and you practice doing it repeatedly from different orientations so it becomes second nature. Depending on your proclivities you may have your ice ax tethered to your hand, so no matter how you fall you can recover it and make use of it, though this can be somewhat dangerous if you lose control of an object with lots of pointy bits and go tumbling down a hill with it guaranteed to be in close proximity to you, so it's a bit of a judgment call. Anywho, if you manage to lose your ax then you can always dig into the snow with your knees and elbows, although this obviously doesn't work so well on hard ice.

Your major other line of defense is your team/partner. If you fall you should yell "FALLING!" and that will trigger everyone in your team to immediately go into an arrest position in the time it takes for the coils of rope to run out. In this case the other climbers will be in a similar position as the self-arrest position with the exception of digging their feet very strongly into the snow/ice as well. This is something that you can do while stationary but not something you want to try while wearing crampons and falling because you'll probably just flip your entire body and/or break your foot. The rope is "dynamic" meaning that it is somewhat elastic, which will moderate the jerking motion of the rope as it becomes taught, which prevents injury to the falling climber but also reduces the force on the anchors (in this case the other climbers). Additionally, when climbing on a cornice like this it might make more sense for the other climber to simply jump to the other side of the ridge line.

If you were climbing over thick ice or right above a steep dropoff with a short runout then it's likely you'd use some form of "protection" gear such as setting an anchor and belaying across or such-like.