  Multigraded linear series generalize the classical morphism to the linear
series of a basepoint-free line bundle on a scheme. We investigate the
collection of the natural cornering morphisms into elementary bigraded linear
series obtained from direct summands of the original globally generated vector
bundle. Our main result is a condition on the injectivity of the product
morphism. We apply our result in three examples: modules over the
reconstruction algebra, equivariant Hilbert and Quot schemes of quotient stacks
and Kapranov's tilting bundle over the Grassmannian.
