  In arrangements of pseudocircles (Jordan curves) the weight of a vertex
(intersection point) is the number of pseudocircles that contain the vertex in
its interior. We give improved upper bounds on the number of vertices of weight
<=k in certain arrangements of pseudocircles in the plane. In particular,
forbidding certain subarrangements we improve the known bound of 6n-12 (cf.
Kedem et al., 1986) for vertices of weight 0 in arrangements of n pseudocircles
to 4n-6. In complete arrangements (i.e. arrangements with each two
pseudocircles intersecting) we identify two subarrangements of three and four
pseudocircles, respectively, whose absence gives improved bounds for vertices
of weight 0 and more generally for vertices of weight <=k.
