[Paper Review] On freeness of divisors on $\mathbb P^2$

Ştefan Tohǎneanu|arXiv (Cornell University)|Mar 9, 2012

Commutative Algebra and Its Applications10 references3 citations

TL;DR

This paper establishes a criterion for freeness of quasihomogeneous divisors in $\mathbb{P}^2$ by showing that the quotient ring $\mathbb{C}[x,y,z]/I$ is arithmetically Cohen-Macaulay—and thus the divisor is free—if and only if there exists a syzygy on the ideal $I$ that forms a regular sequence. The result applies particularly to Jacobian ideals of reduced curves, offering a computational test for freeness via syzygy degrees and enabling bounds on the degree of the reduced Jacobian scheme for free line arrangements.

ABSTRACT

Let $I\subset \mathbb C[x,y,z]$ be an ideal of height 2 and minimally generated by three homogeneous polynomials of the same degree. If $I$ is a locally complete intersection we give a criterion for $\mathbb C[x,y,z]/I$ to be arithmetically Cohen-Macaulay. Since the setup above is most commonly used when $I=J_F$ is the Jacobian ideal of the defining polynomial of a "quasihomogeneous" reduced curve $Y=V(F)$ in $\mathbb P^2$, our main result becomes a criterion for freeness of such divisors. As an application we give an upper bound for the degree of the reduced Jacobian scheme when $Y$ is a free rank 3 central essential arrangement, as well as we investigate the connections between the first syzygies on $J_F$, and the generators of $\sqrt{J_F}$.

Motivation & Objective

• To provide a criterion for freeness of reduced divisors in $\mathbb{P}^2$ defined by quasihomogeneous polynomials.
• To characterize when the quotient ring $\mathbb{C}[x,y,z]/I$ is arithmetically Cohen-Macaulay for height-2, locally complete intersection ideals $I$ minimally generated by three homogeneous polynomials of equal degree.
• To apply the criterion to free hyperplane arrangements in $\mathbb{P}^2$, particularly line arrangements, by linking syzygy degrees to the degree of the reduced Jacobian scheme.
• To investigate the relationship between the degrees of first syzygies on the Jacobian ideal and the degrees of generators of its radical.

Proposed method

• Use of the Hilbert-Burch theorem to analyze the structure of ideals generated by three homogeneous polynomials of equal degree in $\mathbb{C}[x,y,z]$.
• Application of Eisenbud and Huneke's result on ideals with a regular sequence as syzygy to establish the arithmetically Cohen-Macaulay condition.
• Reduction of the freeness problem to checking whether a syzygy of the Jacobian ideal forms a regular sequence in the polynomial ring.
• Use of the linear type property (equivalent to local complete intersection for height-2 ideals) to ensure the ideal is well-behaved for resolution analysis.
• Computation of bounds on $\deg(\sqrt{J_F})$ using syzygy degrees and the structure of free arrangements.
• Construction of examples and counterexamples to test the sharpness of bounds and the necessity of assumptions, including Example 2.4 showing the criterion fails in higher dimensions.

Experimental results

Research questions

• RQ1 When is the quotient ring $\mathbb{C}[x,y,z]/I$ arithmetically Cohen-Macaulay for a height-2, locally complete intersection ideal $I$ minimally generated by three homogeneous polynomials of equal degree?
• RQ2 What conditions on the syzygies of the Jacobian ideal $J_F$ imply that the curve $V(F) \subset \mathbb{P}^2$ is free?
• RQ3 Can the degree of the reduced Jacobian scheme of a free line arrangement in $\mathbb{P}^2$ be bounded in terms of syzygy degrees?
• RQ4 How do the degrees of the first syzygies on $J_F$ relate to the degrees of the generators of $\sqrt{J_F}$ for transverse arrangements of smooth curves?
• RQ5 Is the upper bound $\deg(\sqrt{J_F}) \leq b^2 + b + 1$ for free arrangements with exponents $\{1,a,b\}$ sharp, and can it be improved?

Key findings

• The ring $\mathbb{C}[x,y,z]/I$ is arithmetically Cohen-Macaulay if and only if there exists a syzygy on $I$ that forms a regular sequence.
• For a free line arrangement $\mathcal{A}$ of $n$ lines not of the pencil-plus-generic-line type, $\deg(\sqrt{J_Q}) \leq (n-3)^2 + (n-3) + 1 = n^2 - 5n + 7$, and if $\deg(\sqrt{J_Q}) \geq n^2 - 5n + 8$, then $\mathcal{A}$ is not free.
• The bound $\deg(\sqrt{J_F}) \leq b^2 + b + 1$ holds for free arrangements with exponents $\{1,a,b\}$, $a \leq b$, and this bound is sharp in some cases.
• For transverse arrangements of smooth curves, $\alpha(\sqrt{J_F}) \leq \beta(J_F) + 1$, where $\beta(J_F)$ is the minimal degree of a syzygy on $J_F$, and this bound is attained in Example 3.4.
• The reduced Jacobian scheme of a free divisor $Y = V(F)$ in $\mathbb{P}^2$ satisfies $V(J_F) = V(I^{(A_1,B_1,C_1)}) \cap V(I^{(A_2,B_2,C_2)})$ when $\{ (A_1,B_1,C_1), (A_2,B_2,C_2) \}$ is a basis for the first syzygy module.
• Example 3.4 shows a case where $\beta(J_Q) = 2$ and $\alpha(\sqrt{J_Q}) = 3$, confirming the sharpness of the bound $\alpha(\sqrt{J_F}) \leq \beta(J_F) + 1$.