  We characterize by pattern avoidance the Schubert varieties for GL_n which
are local complete intersections (lci). For those Schubert varieties which are
local complete intersections, we give an explicit minimal set of equations
cutting out their neighborhoods at the identity. Although the statement of our
characterization only requires ordinary pattern avoidance, showing that the
Schubert varieties not satisfying our conditions are not lci appears to require
working with more general notions of pattern avoidance. The Schubert varieties
defined by inclusions, originally introduced by Gasharov and Reiner, turn out
to be an important subclass, and we further develop some of their
combinatorics. Applications include formulas for Kostant polynomials and
presentations of cohomology rings for lci Schubert varieties.
