  We consider the coordinate-space matrix elements that correspond to
fixed-angle scattering amplitudes involving partons and Wilson lines in
coordinate space, working in Feynman gauge. In coordinate space, both collinear
and short-distance limits produce ultraviolet divergences. We classify
singularities in coordinate space, and identify neighborhoods associated
unambiguously with individual subspaces (pinch surfaces) where the integrals
are singular. The set of such regions is finite for any diagram. Within each of
these regions, coordinate-space soft-collinear and hard-collinear
approximations reproduce singular behavior. Based on this classification of
regions and approximations, we develop a series of nested subtraction
approximations by analogy to the formalism in momentum space. This enables us
to rewrite each amplitude as a sum of terms to which gauge theory Ward
identities can be applied, factorizing them into hard, jet and soft factors,
and to confirm the multiplicative renormalizability of products of lightlike
Wilson lines. We study in some detail the simplest case, the color-singlet cusp
linking two Wilson lines, and show that the logarithm of this amplitude, which
is a sum of diagrams known as webs, is closely related to the corresponding
subtracted amplitude order by order in perturbation theory. This enables us to
confirm that the logarithm of the cusp can be written as the integral of an
ultraviolet-finite function over a surface. We study to what extent this result
generalizes to amplitudes involving multiple Wilson lines.
