The projective hull of certain curves in ℂ ²

The projective hull X̌ of a compact set X ⊂ ℙ ⁿ is an analogue of the classical polynomial hull of a set in ℂ ⁿ. In the special case that X ⊂ ℂ ⁿ ⊂ ℙ ⁿ, the affine part X̌ ∩ ℂ ⁿ can be defined as the set of points x ∈ ℂ ⁿ for which there exists a constant M _x so that |p(x)| ≤ M _x ^d sup _X |p| for all polynomials p of degree ≤ d, and any d ≥ 1. Let X̌ (M) be the set of points x where M _x can be chosen ≤ M. Using an argument of E. Bishop, we show that if γ ⊂ ℂ ² is a compact real analytic curve (not necessarily connected), then for any linear projection π : ℂ ² → ℂ, the set γ̌(M) ∩ π ^-1(z) is finite for almost all z ∈ ℂ It is then shown that for any compact stable real-analytic curve γ ⊂ ℙ ⁿ, the set γ̌ - γ̌ is a 1-dimensional complex analytic subvariety of ℙ ⁿ -γ. Boundary regularity for γ̌ is also discussed in detail.