A full intrinsic quadric is a normal complete variety with a finitely generated Cox ring defined by a single quadratic relation of full rank. We describe all surfaces of this type explicitly via local Gorenstein indices. As applications, we present upper and lower bounds in terms of the Gorenstein index for the degree, the log canonicity and the Picard index. Moreover, we determine all full intrinsic quadric surfaces admitting a Kahler-Einstein metric.

We are grateful to the referee for carefully reading the manuscript and for providing us with helpful comments and valuable suggestions.