[Paper Review] Syzygies of Jacobian ideals and defects of linear systems

Alexandru Dimca|arXiv (Cornell University)|Oct 5, 2012

Advanced Differential Equations and Dynamical Systems6 references20 citations

TL;DR

This paper establishes a precise connection between the syzygies of the partial derivatives of a homogeneous polynomial and the defect of linear systems vanishing on the singular locus of the corresponding hypersurface in projective space. Using the Cayley-Bacharach Theorem and properties of the Milnor and Tjurina algebras, it proves that the saturation of the Jacobian ideal stabilizes at $\max(T - ct(D), st(D))$, where $T = (n+1)(d-2)$, and derives explicit formulas for the $a$-invariant and Castelnuovo-Mumford regularity of the Milnor algebra.

ABSTRACT

Our main result describes the relation between the syzygies involving the first order partial derivatives $f_0,...,f_n$ of a homogeneous polynomial $f\in \C[x_0,...x_n]$ and the defect of the linear systems vanishing on the singular locus subscheme $Σ_f=V(f_0,...,f_n)$ of the hypersurface $D:f=0$ in the complex projective space $\PP^n$, when $D$ has only isolated singularities.

Motivation & Objective

• To understand the defect of linear systems vanishing on the singular locus scheme $\Sigma_f = V(f_0, \dots, f_n)$ of a projective hypersurface $D:f=0$ with isolated singularities.
• To clarify the relationship between syzygies of the partial derivatives $f_0, \dots, f_n$ and the saturation of the Jacobian ideal $J_f$.
• To determine the exact stabilization threshold for the saturation $\widehat{J}_f$ in terms of invariants $ct(D)$ and $st(D)$.
• To derive explicit formulas for the $a$-invariant and Castelnuovo-Mumford regularity of the Milnor algebra $M(f)$.
• To extend results from the nodal case to general isolated singularities using the full power of the Cayley-Bacharach Theorem.

Proposed method

• Uses the Koszul complex $K^*(f)$ to define the minimal degree of a nontrivial syzygy, $mdr(D)$, via $H^n(K^*(f))_{q+n} \neq 0$.
• Applies the Cayley-Bacharuch Theorem (CB7) to relate residual subschemes and syzygy structures in the context of the singular locus $\Sigma_f$.
• Relies on the Hilbert-Poincaré series $HP(M(f); t)$ and compares it to that of a smooth hypersurface $M(f_s)$, using $HP(M(f_s); t) = \frac{(1-t^{d-1})^{n+1}}{(1-t)^{n+1}}$.
• Defines key invariants: $ct(D)$ as the coincidence threshold, $st(D)$ as the stability threshold, and $mdr(D)$ as the minimal degree of syzygy.
• Uses the relation $ct(D) = mdr(D) + d - 2$ to express saturation bounds in terms of $ct(D)$ and $st(D)$.
• Applies the saturation condition $\widehat{J}_f = \{ s \in S \mid \exists m_i \text{ such that } x_i^{m_i}s \in J_f \}$ to determine when $J_f$ becomes saturated.

Experimental results

Research questions

• RQ1 How do syzygies among the partial derivatives $f_0, \dots, f_n$ relate to the defect of linear systems vanishing on the singular locus $\Sigma_f$?
• RQ2 What is the precise stabilization threshold for the saturation $\widehat{J}_f$ of the Jacobian ideal in terms of $ct(D)$ and $st(D)$?
• RQ3 How do the invariants $ct(D)$, $st(D)$, and $mdr(D)$ control the structure of the Milnor algebra $M(f)$ and its $a$-invariant and regularity?
• RQ4 Under what conditions does the saturation $\widehat{J}_f$ stabilize exactly at $st(D)$?
• RQ5 Can the Hilbert-Poincaré series of $M(f)$ be expressed in terms of that of $M(f_s)$ and the structure of the singular locus when $\Sigma_f$ is a complete intersection?

Key findings

• The saturation $\widehat{J}_f$ satisfies $\widehat{J}_{f,k} = J_{f,k}$ for all $k \geq \max(T - ct(D), st(D))$, where $T = (n+1)(d-2)$.
• The $a$-invariant of the Milnor algebra is given by $a(M(f)) = T - ct(D) - 1$.
• The Castelnuovo-Mumford regularity of $M(f)$ is $\operatorname{reg}(M(f)) = \max(T - ct(D), \operatorname{sat}(J_f) - 1)$.
• When $\Sigma_f$ is a complete intersection defined by forms of degrees $a_1, \dots, a_n$, then $\tau(D) = a_1 \cdots a_n$ and $ct(D) = T - \sum a_i + n$.
• For nodal hypersurfaces in $\mathbb{P}^n$, $ct(D) \geq T/2$, and $st(D) = 2d - 3$ for $n=2$, leading to $\operatorname{sat}(J_f) = st(D)$ under certain conditions.
• When $ct(D) \geq T/2$, the Tjurina number $\tau(D)$ is bounded by $\dim M(f_s)_{T - ct(D)}$, implying that large $ct(D)$ forces small $\tau(D)$.