  In this paper we prove a direct geometric relation between Deligne--Lusztig
varieties and Springer fibres in type $\mathsf{A}$: For any rational unipotent
element, the Springer fibre cuts out a unique component of a specific
Deligne--Lusztig variety; moreover, this component forms an open dense subset
of a component of the Springer fibre. This boils down to a map from the
unipotent variety to the Weyl group, and combines several constructions with a
combinatorial flavour (like Weyr normal forms, Robinson--Schensted
correspondence, and Spaltenstein's and Steinberg's labellings); it also
provides a geometric interpretation of a classical dimension formula of
unipotent centralisers.
