[Paper Review] Holomorphic curves into algebraic varieties intersecting moving hypersurface targets

Gerd Dethloff, Van Tan Tran|arXiv (Cornell University)|Mar 30, 2015

Meromorphic and Entire Functions24 references20 citations

TL;DR

This paper establishes a Second Main Theorem for algebraically nondegenerate holomorphic curves in complex projective varieties intersecting moving hypersurface targets, extending Ru's fixed-target result. By introducing a novel filtration method on the coordinate ring of the variety and analyzing Hilbert sequence asymptotics, the authors overcome the failure of regular sequences in general varieties and prove that counting functions satisfy a sharp inequality with error term o(T_f(r)).

ABSTRACT

27 pages. arXiv: 1503.08801v2. We correct an error in the proof: The second part of equation (2.12) in the the first version of this paper does not hold in general.

Motivation & Objective

• To generalize Min Ru's Second Main Theorem for fixed hypersurfaces to the case of moving hypersurface targets in projective varieties.
• To overcome the breakdown of the regular sequence property in general varieties, which invalidates prior methods relying on linear isomorphisms from regular sequences.
• To develop a new filtration technique on the homogeneous coordinate ring of the variety that preserves dimension control despite the absence of regular sequences.
• To control the locus where moving hypersurfaces fail to be in general position using an element of the inertia ideal, even when the ideal is not principal.
• To establish a sharp counting function inequality for holomorphic curves with error term o(T_f(r)), extending results to arbitrary projective varieties without completeness or Cohen-Macaulay assumptions.

Proposed method

• Construct a filtration on the homogeneous coordinate ring of the projective variety V, using the moving hypersurfaces' defining polynomials.
• Analyze the Hilbert sequence asymptotics of the filtration to compute the sum of dimensions of factor vector spaces, leveraging combinatorial techniques.
• Prove that almost all factor vector spaces in the filtration have the same dimension, enabling precise asymptotic estimates.
• Use a generic specialization argument to show that vector space dimensions remain stable under evaluation of meromorphic coefficients at generic points.
• Introduce a control element from the inertia ideal of the moving hypersurfaces to manage the non-general-position locus, even when the ideal is not principal.
• Apply a generalized version of Mumford's identity via Evertse-Ferretti estimates, combined with maximal function estimates on the logarithmic ratio of norms.

Experimental results

Research questions

• RQ1 Can the Second Main Theorem for holomorphic curves in projective varieties be extended from fixed to moving hypersurface targets?
• RQ2 How can one maintain dimension control in filtrations of the coordinate ring when the regular sequence property fails?
• RQ3 What is the correct way to handle the non-transversality locus where moving hypersurfaces are not in general position?
• RQ4 Can the error term in the counting function inequality be reduced to o(T_f(r)) without assuming the variety is a complete intersection?
• RQ5 Is it possible to generalize the method of Evertse-Ferretti and Ru to moving targets using only algebraic and asymptotic techniques on Hilbert functions?

Key findings

• The authors prove a Second Main Theorem for algebraically nondegenerate holomorphic curves f: ℂ → ℙ^M intersecting moving hypersurfaces, with the counting function satisfying ∑_{j=1}^q N_f(r, Q_j) ≥ d(q−n−1−ε)T_f(r) + o(T_f(r)) for any ε > 0.
• The method successfully avoids reliance on the regular sequence property by using Hilbert sequence asymptotics and combinatorial dimension control in the filtration.
• The dimension of the factor vector spaces in the filtration is shown to be almost constant, allowing precise asymptotic estimates even when the variety is not Cohen-Macaulay.
• The locus where moving hypersurfaces fail to be in general position is controlled via a single element of the inertia ideal, which suffices for the error term analysis.
• The proof establishes that the error term is o(T_f(r)), confirming the sharpness of the bound under the given assumptions.
• The result generalizes Ru's fixed-target Second Main Theorem and Dethloff-Tan's moving-target result in ℙ^n to arbitrary projective varieties without requiring the variety to be a complete intersection.