We study the qualitative structure of the set LS of integral L-space surgery slopes for links with two components. It is known that the set of L-space surgery slopes for a nontrivial L-space knot is always a positive half-line. However, already for two-component torus links the set LS has a very complicated structure, e.g. in some cases it can be unbounded from below. In order to probe the geometry of this set, we ask if it is bounded from below for more general L-space links with two components. For algebraic two-component links we provide three complete characterizations for the boundedness from below: one in terms of the Alexander polynomial, one in terms of the embedded resolution graph, and one in terms of the so-called h-function introduced by the authors in [9]. It turns out that LS is bounded from below for most algebraic links. If it is unbounded from below, it must contain a negative half-line parallel to one of the axes. We also give a sufficient condition for boundedness for arbitrary L-space links.