On the Nochka-chen-ru-wong Proof of Cartan’s Conjecture

In 1982–83, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in P relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his 1987 thesis, and subseqently simplified by M. Ru and P.-M. Wong in 1991. The proof involved assigning weights to the hyperplanes. This paper provides a mild simplification of the proof of the construction of the weights, by construing the “Nochka diagram” of Ru and Wong as a convex hull. In 1982 and 1983, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in P relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his thesis [C], and subsequently simplified by M. Ru and P.-M. Wong [R-W]. This note offers some further simplifications, consisting of defining the Nochka polygon in terms of a convex hull (described in a previous letter [V 1]), and rewording the combinatorics to use linear subspaces of P instead of sets of hyperplanes (motivated by the rephrasing of ([R-W], Thm. 2.2) in [S]). This note only addresses the proof of the existence of the Nochka weights (Theorem 3). For details on the remainder of Nochka’s proof, see [R-W]; simplified versions are also given in [S] and [V 2]. Let H1, . . . ,Hq be hyperplanes in P k , not necessarily distinct, but in n-subgeneral position; i.e., there exists an embedding of P as a linear subspace of P and (distinct) hyperplanes H ′ 1, . . . ,H ′ q in general position in P n such that Hi = H ′ i ∩ P k for all i . For linear subspaces L ⊆ P , define α(L) = #{i : Hi ⊇ L} and recall that codimL denotes the codimension of L in P ; i.e., codimL = k−dimL . Also, by convention, let codim ∅ = k + 1 . For linear subspaces L ⊆ P let P (L) = (α(L), codimL) ∈ R . Supported by NSF grants DMS-9304899, DMS-0200892, and DMS-0500512. 1