This paper deals with the conditions under which singular definites, on the one hand, and universally quantified DPs, on the other hand, receive interpretations according to which the sets denoted by the NP-complements of the respective determiners vary with the situations quantified over by a Q-adverb. First, it is shown that in both cases such interpretations depend on the (contextual) availability of a situation predicate that characterizes a set of situations such that each of those situations contains exactly one individual (in the case of singular definites) or at least two individuals (in the case of universally quantified DPs) that satisfy the respective NP-predicate. It is argued that these constraints are due to the presuppositions associated with the respective determiner. Secondly, it is shown that there is an important difference between the two types of sentences: in the case of adverbially quantified sentences with universally quantified DPs, the Q-adverb has to c-command the DP overtly in order for co-variation to be possible. In the case of adverbially quantified sentences with singular definites, on the other hand, this is not necessary. This difference is accounted for in the following way: in both cases, the Q-adverb has to c-command the DP at LF in order to bind the situation variable contained within the NP-complement of the respective determiner. Furthermore, only focal DPs can be reconstructed at LF, and co-varying definites are novel definites in the sense of Umbach (2001) and therefore have to be focus marked. This, however, is not the case for universally quantified DPs, as they are unable to introduce discourse referents anyway. The difference between singular definites and universally quantified DPs is thus due to the fact that only the former can be reconstructed in the type of sentence under discussion.