  A Poisson pencil is called flat if all brackets of the pencil can be
simultaneously locally brought to a constant form. Given a Poisson pencil on a
3-manifold, we study under which conditions it is flat. Since the works of
Gelfand and Zakharevich, it is known that a pencil is flat if and only if the
associated Veronese web is trivial. We suggest a simpler obstruction to
flatness, which we call the curvature form of a Poisson pencil. This form can
be defined in two ways: either via the Blaschke curvature form of the
associated web, or via the Ricci tensor of a connection compatible with the
pencil. We show that the curvature form of a Poisson pencil can be given by a
simple explicit formula. This allows us to study flatness of linear pencils on
three-dimensional Lie algebras, in particular those related to the argument
translation method. Many of them appear to be non-flat.
