The Projective Hull of Certain Curves in C

The projective hull X̂ of a compact set X ⊂ P is an analogue of the classical polynomial hull of a set in C. In the special case that X ⊂ C ⊂ P, the affine part X̂ ∩C can be defined as the set of points x ∈ C for which there exists a constant Mx so that |p(x)| ≤ M x sup X |p| for all polynomials p of degree ≤ d, and any d ≥ 1. Let X̂(M) be the set of points x where Mx can be chosen ≤ M . Using an argument of E. Bishop, we show that if γ ⊂ C is a compact real analytic curve (not necessarily connected), then for any linear projection π : C → C, the set γ̂(M) ∩ π(z) is finite for almost all z ∈ C. It is then shown that for any compact stable real-analytic curve γ ⊂ P, the set γ̂−γ is a 1-dimensional complex analytic subvariety of P − γ. Partially supported by the N.S.F.