[Paper Review] Nearly free divisors and rational cuspidal curves

Alexandru Dimca, Gabriel Sticlaru|arXiv (Cornell University)|May 4, 2015

Algebraic Geometry and Number Theory18 references22 citations

TL;DR

This paper introduces nearly free divisors—curves whose Jacobian algebras have a minimal resolution slightly more complex than free divisors—and proves that all rational cuspidal curves of even degree, prime power degree, or with abelian fundamental group are either free or nearly free. Using Walther's recent result on local cohomology, the authors establish a strong structural classification for these curves, extending previous freeness results and supporting a broader conjecture on the freeness class of rational cuspidal curves.

ABSTRACT

We define a class of plane curves which are close to the free divisors and such that conjecturally it contains the class of rational cuspidal curves. Using a recent result by U. Walther we show that any unicuspidal rational curve with a unique Puiseux pair is either free or belongs to this class.

Motivation & Objective

• To define a new class of plane curves—nearly free divisors—that are close to free divisors in terms of their Jacobian algebra's local cohomology and resolution structure.
• To prove that all rational cuspidal curves of even degree, prime power degree, or with abelian fundamental group are either free or nearly free, supporting a broader conjecture on their classification.
• To extend the understanding of the freeness property in rational cuspidal curves by analyzing the dimensions of the Milnor algebra components and their saturation with respect to the maximal ideal.
• To provide a structural characterization of nearly free curves via their exponents and resolution type, generalizing known results on free curves.

Proposed method

• Define nearly free divisors via the condition that the local cohomology module $ N(f) = I_f / J_f $ satisfies $ \dim N(f)_k \leq 1 $ for all $ k $, generalizing the freeness condition $ N(f) = 0 $.
• Use the graded dimensions $ m(f)_k = \dim M(f)_k $ of the Milnor algebra $ M(f) = S / J_f $ to characterize freeness and nearly freeness.
• Apply Uli Walther's recent result on the structure of local cohomology modules to prove that rational cuspidal curves of even degree or prime power degree are either free or nearly free.
• Construct explicit resolutions of the Milnor algebra for nearly free curves using three generators $ r_1, r_2, r_3 $ of degrees $ d_1, d_2, d_3 $ with $ d_2 = d_3 $, and a syzygy $ R $.
• Analyze the Tjurina number $ \tau(C) $, total Tjurina number, and other invariants in terms of the exponents $ (d_1, d_2) $, showing $ d_1 + d_2 = d $ and $ \tau(C) = 3d(d-2)/4 $ for certain families.
• Verify the nearly freeness of known rational cuspidal curves (e.g., from Sakai-Tono classification) by computing their resolution type and checking $ \dim N(f)_k \leq 1 $.

Experimental results

Research questions

• RQ1 Are all rational cuspidal curves in the plane either free or nearly free, as conjectured?
• RQ2 Can the freeness or nearly freeness of a rational cuspidal curve be determined solely by the dimensions $ m(f)_k $ of its Milnor algebra?
• RQ3 Does the structure of the local cohomology module $ N(f) = I_f / J_f $ fully characterize the nearly free property?
• RQ4 For which degrees and fundamental group types can the conjecture that rational cuspidal curves are free or nearly free be proven?
• RQ5 What is the precise relationship between the exponents $ (d_1, d_2, d_3) $ of the resolution and the geometric invariants like $ \tau(C) $, $ ct $, and $ st $?

Key findings

• All rational cuspidal curves of even degree are either free or nearly free, as proven using Walther's result on local cohomology.
• The conjecture holds for rational cuspidal curves of prime power degree or with abelian fundamental group, as shown in Corollary 4.2.
• For unicuspidal curves with a single Puiseux pair, the conjecture holds except in one odd-degree case where topological assumptions fail.
• The family of curves $ C_d: f_d = (y^k z + x^{k+1})^2 - x y^{2k+1} $ of even degree $ d = 2k+2 $ is almost free with exponents $ (k+1, k+1, k+1) $ and $ \tau = 3k(k+1) $.
• The curves $ C_{j,k}: f = (y^{k+j} z + x^{k+j+1})^2 - x^{2j+1} y^{2k+1} $ of even degree $ d = 2(k+j)+2 $ are almost free with $ d_1 = d_2 = d_3 = k+j+1 $, $ \tau = 3d(d-2)/4 $, and $ ct = st = (3d-4)/2 $.
• The resolution of the Milnor algebra for nearly free curves has the form $ 0 \to S(-d_1 - d) \oplus S(-d_2 - d)^2 \to S^3(-d+1) \to S \to 0 $, with $ d_2 = d_3 $, generalizing the free case.