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This article studies covers in $\PG(3,q)$ and in generalized quadrangles. The excess of a cover is defined to be the difference between the number of lines in the cover and the number of lines in a spread. In contrast with the theory of partial spreads which tells us that large partial spreads can be extended to spreads, in $\PG(3,q)$ and in some generalized quadrangles, there exist minimal covers with small excess. For such minimal covers with small excess, we describe the structure of the set of points lying on at least two lines of the cover.

Covers of {${\rm PG}(3,q)$} and of finite generalized quadrangles