  The transverse-field $XY$ chain with the long-range interactions was
investigated by means of the exact-diagonalization method. The algebraic decay
rate $\sigma$ of the long-range interaction is related to the effective
dimensionality $D(\sigma)$, which governs the criticality of the
transverse-field-driven phase transition at $H=H_c$. According to the large-$N$
analysis, the phase boundary $H_c(\eta)$ exhibits a reentrant behavior within
$2<D < 3.065\dots $, as the $XY$-anisotropy $\eta$ changes. On the one hand, as
for the $D=(2+1)$ and $(1+1)$ short-range $XY$ magnets, the singularities have
been determined as $H_c(\eta) -H_c(0) \sim |\eta|$ and $ 0 $, respectively, and
the transient behavior around $D \approx 2.5$ remains unclear. As a preliminary
survey, setting $(\sigma,\eta)=(1 ,0.5)$, we investigate the phase transition
by the agency of the fidelity, which seems to detect the singularity at $H=H_c$
rather sensitively. Thereby, under the setting $ \sigma=4/3$ ($D=2.5$), we cast
the fidelity data into the crossover-scaling formula with the properly scaled
$\eta$, aiming to determine the multi-criticality around $\eta=0$. Our result
indicates that the multi-criticality is identical to that of the $D=(2+1)$
magnet, and $H_c(\eta)$'s linearity might be retained down to $D>2$.
