  Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet
is conjectured to be the canonical scaling limit for last passage percolation
models. In recent work arXiv:1812.00309 of Dauvergne, Ortmann, and Vir\'ag,
this object was constructed and shown to be the limit after parabolic
correction of one such model: Brownian last passage percolation. This limit
object, called the directed landscape, admits geodesic paths between any two
space-time points $(x,s)$ and $(y,t)$ with $s<t$. In this article, we examine
fractal properties of the set of these paths. Our main results concern
exceptional endpoints admitting disjoint geodesics. First, we fix two distinct
starting locations $x_1$ and $x_2$, and consider geodesics traveling
$(x_1,0)\to (y,1)$ and $(x_2,0)\to (y,1)$. We prove that the set of
$y\in\mathbb{R}$ for which these geodesics coalesce only at time $1$ has
Hausdorff dimension one-half. Second, we consider endpoints $(x,0)$ and $(y,1)$
between which there exist two geodesics intersecting only at times $0$ and $1$.
We prove that the set of such $(x,y)\in\mathbb{R}^2$ also has Hausdorff
dimension one-half. The proofs require several inputs of independent interest,
including (i) connections to the so-called difference weight profile studied in
arXiv:1904.01717; and (ii) a tail estimate on the number of disjoint geodesics
starting and ending in small intervals. The latter result extends the analogous
estimate proved for the prelimiting model in arXiv:1709.04110.
