The Projective Hull of Certain Curves in C^2

TL;DR

This paper investigates the projective hulls of certain curves in complex projective space, proving a conjecture that relates the hulls of stable real-analytic curves to complex analytic subvarieties.

Contribution

It establishes a new result connecting the projective hulls of stable real-analytic curves in P^n to complex analytic subvarieties, confirming a conjecture in the field.

Findings

The set of points in the hull above a generic point is finite.

The hull of a stable real-analytic curve forms a 1-dimensional complex analytic subvariety.

Proves the conjecture relating hulls and complex subvarieties for stable curves.

Abstract

The projective hull X^ of a subset X in complex projective space P^n is an analogue of the classical polynomial hull of a set in C^n. If X is contained in an affine chart C^n on P^n, then the affine part of X^ is the set of points x in C^n for which there exists a constant M=M_x so that |p(x)| < M^d sup{|p(y)| : y in X} for all polynomials p of degree less than or equal to d, and any d > 0. Let X^(M) be the set of points x where M_x can be chosen < M. Using an argument of E. Bishop, we show the following. Let G be a compact real analytic curve (not necessarily connected) in C^2. Then for any linear projection p: C^2 --> C^1, the set of points in G^(M) lying above a point z in C^1 is finite for almost all z. Using this, we prove the conjecture that for any compact stable real-analytic curve G in P^n, the set G^-G is a 1-dimensional complex analytic subvariety of P^n-G.