  Vertex operators, being families of birational transformations of
infinite-dimensional algebraic ``varieties'' M, act on appropriate line bundles
on M. However, they act on (meromorphic) sections only as_partial operators_:
they are defined on a subspace (in an appropriate_lattice of subspaces_ of
Mer(M)), and send "smooth families" of vectors in such a subspace to smooth
families. Axiomatizing this, we define_conformal fields_ as arbitrary families
of partial operators in Mer(M) which satisfy both these properties.
  The ``variety'' M related to standard vertex operators is formed by rational
functions of one variable z in Z=P^1, changing the variety Z one obtains
different examples of M, and multidimensional analogues of vertex operators.
One can cover Mer(M) by ``big smooth subsets''; these subsets are parameterized
by appropriate projective bundles over Hilbert schemes of points on Z. We
deduce conformal associativity relation for conformal fields, and conformal
commutation relations for Laurent coefficients of commuting conformal fields
from geometric properties of the Hilbert schemes.
  We start with providing examples of OPE in smooth families, i.e., smooth
families a(s) and b(t) (of partial linear operators) such that a(s)b(t) has a
pole when s=t. We also provide geometric description of boson-fermion
correspondence, and relations of this correspondence to geometry of the set of
meromorphic functions.
