  We give a geometric construction of the W_{1+infty} vertex algebra as the
infinitesimal form of a factorization structure on an adelic Grassmannian. This
gives a concise interpretation of the higher symmetries and Backlund-Darboux
transformations for the KP hierarchy and its multicomponent extensions in terms
of a version of "W_{1+infty}-geometry": the geometry of D-bundles on smooth
curves, or equivalently torsion-free sheaves on cuspidal curves.
