  We construct geometrically the generating fields of a W algebra which acts
irreducibly on the direct sum of the cohomology rings of the Hilbert schemes of
n points on a projective surface for all n. We compute explicitly the
commutators among a set of linear basis elements of the W algebra, and identify
this algebra with a $W_{1+\infty}$-type algebra. A precise formula of certain
Chern character operators, which is essential for the construction of the W
algebra, is established in terms of the Heisenberg algebra generators. In
addition, these Chern character operators are proved to be the zero-modes of
vertex operators.
