  We present a study of the stellar populations of globular clusters (GCs) in
the Virgo Cluster core with a homogeneous spectroscopic catalog of 692 GCs
within a major axis distance $R_{\rm maj} = $ 840 kpc from M87. We investigate
radial and azimuthal variations in the mean age, total metallicity, [Fe/H], and
$\alpha$-element abundance, of blue (metal-poor) and red (metal-rich) GCs using
their co-added spectra. We find that the blue GCs have a steep radial gradient
in [Z/H] within $R_{\rm maj} =$ 165 kpc, with roughly equal contributions from
[Fe/H] and [$\alpha$/Fe], and flat gradients beyond. By contrast, the red GCs
show a much shallower gradient in [Z/H], which is entirely driven by [Fe/H]. We
use GC-tagged Illustris simulations to demonstrate an accretion scenario where
more massive satellites (with more metal- and $\alpha$-rich GCs) sink further
into the central galaxy than less massive ones, and where the gradient
flattening occurs because of the low GC occupation fraction of low-mass dwarfs
disrupted at larger distances. The dense environment around M87 may also cause
the steep [$\alpha$/Fe] gradient of the blue GCs, mirroring what is seen in the
dwarf galaxy population. The progenitors of red GCs have a narrower mass range
than those of blue GCs, which makes their gradients shallower. We also explore
spatial inhomogeneity in GC abundances, finding that the red GCs to the
northwest of M87 are slightly more metal-rich. Future observations of GC
stellar population gradients will be useful diagnostics of halo merger
histories.
