STUDIA MATHEMATICA 165 (1) (2004) Polynomially convex hulls of families of arcs by Zbigniew Slodkowski (Chicago, IL) Abstract. The paper is devoted to the study of polynomially convex hulls of compact subsets of C 2 , fibered over the boundary of the unit disc, such that all fibers are simple arcs in the plane and their endpoints form boundaries of two closed, not intersecting analytic discs. The principal question concerned is under what additional condition such a hull is a bordered topological hypersurface and, in particular, is foliated by a unique holomorphic motion. One of the main results asserts that this happens when the family of arcs satisfies the Continuous Cone Condition. 0. Motivation. The subject of this paper has originated from the fol- lowing result of the author. (Throughout the paper, D will denote the open unit disc in the complex plane C, centered at 0; D(a,R) will stand for the disc of radius R, centered at a.) Theorem 0.1 ([Sl 1]). Let X ⊂ ∂D × C be a compact set with all fibers X ζ = {w ∈ C :(ζ,w) ∈ X }, ζ ∈ ∂D, nonempty , connected and simply connected. Let Y :=  X (the polynomial hull of X ) and S := the topological boundary of Y \ X relative to D × C. Then either Y = X or Y \ X can be represented as union of a family of analytic discs (= graphs of bounded analytic functions f : D → C). Furthermore S has a unique foliation by such analytic discs. The problems considered in this paper center around the following ques- tion. Assuming all X ζ ’s are simple arcs, when is Y =  X a topological hyper- surface? In case X is a C 2 -regular surface this is a consequence of Forstneric [F]. On the other hand, Example 5.1 below shows that the hull of a family of arcs may have a nonempty interior. In view of these observations the task we undertake in this paper is that of finding minimal regularity assumptions for the hull Y to be a topological hypersurface. Apart from the intrinsic interest of this question for the study of poly- nomial hulls, our motivation comes also from the possibility of applications to constructions of holomorphic motions as suggested by [Sl 3]. The method 2000 Mathematics Subject Classification : Primary 32E20; Secondary 32A37. [1]