[Paper Review] Logarithmic vector fields along smooth plane cubic curves

Kazushi Ueda, Masahiko Yoshinaga|ArXiv.org|Oct 10, 2007

Advanced Differential Equations and Dynamical Systems5 references7 citations

TL;DR

This paper establishes a Torelli-type theorem for smooth plane cubic curves by showing that the sheaf of logarithmic vector fields along a cubic curve determines the curve uniquely if and only if its $j$-invariant is non-zero. The proof links the jumping lines of the sheaf to the Cayleyan curve of the cubic, and uses the structure of Hesse forms and cohomological duality to show that non-vanishing $j$-invariant ensures reconstruction of the curve from the sheaf, while vanishing $j$-invariant leads to non-Torelli examples with isomorphic sheaves along non-isomorphic curves.

ABSTRACT

We study the sheaves of logarithmic vector fields along smooth cubic curves in the projective plane, and prove a Torelli-type theorem in the sense of Dolgachev-Kapranov for those with non-vanishing j-invariants.

Motivation & Objective

• To determine when the sheaf of logarithmic vector fields along a smooth plane cubic curve determines the curve uniquely (Torelli property).
• To analyze the role of the $j$-invariant in distinguishing curves via their logarithmic sheaves.
• To establish a Torelli-type result for smooth cubic curves in $\mathbb{P}^2$ using the geometry of jumping lines and the Cayleyan curve.
• To identify conditions under which the sheaf $\mathcal{T}(-\log D)$ fails to reconstruct $D$, particularly when $j(D) = 0$.

Proposed method

• Use de Rham–Saito’s lemma to analyze the cohomology of the complex $df \wedge \cdot$ on the module of differential forms.
• Identify $D_0(-\log f)$ with the kernel of $df \wedge$ on $\Omega^2$, leading to a free resolution of the logarithmic vector fields.
• Characterize the set of jumping lines of $\mathcal{T}(-\log D)$ as the Cayleyan curve of the cubic $D$, using the condition that $a_i \partial_i f \in \alpha \cdot V^*$.
• Express the Cayleyan curve as a Hesse-type cubic in the dual plane, showing it is smooth iff $j(D) \neq 0$, and singular iff $j(D) = 0$.
• Use restriction of the logarithmic sheaf to another cubic $E$ to relate $H^0(\mathcal{F}|_E)$ to membership of $g$ in the degree-3 part of the Jacobi ideal $J(f)_3$, via duality.
• Apply explicit computation in Hesse form to show that $J(f_t)_3$ is defined by the linear condition $a_{012} + t(a_{000} + a_{111} + a_{222}) = 0$, enabling comparison of curves with isomorphic sheaves.

Experimental results

Research questions

• RQ1 When does the sheaf $\mathcal{T}(-\log D)$ of logarithmic vector fields along a smooth plane cubic $D$ determine $D$ up to isomorphism?
• RQ2 How is the $j$-invariant of a smooth cubic curve related to the geometry of its Cayleyan curve and the jumping lines of $\mathcal{T}(-\log D)$?
• RQ3 Can the Torelli property fail for smooth plane cubics, and if so, under what conditions?
• RQ4 What is the structure of the sheaf $\mathcal{T}(-\log D)$ when $j(D) = 0$, and how does it affect curve reconstruction?
• RQ5 Do non-isomorphic smooth cubic curves exist with isomorphic sheaves $\mathcal{T}(-\log D)$?

Key findings

• The sheaf $\mathcal{T}(-\log D)$ determines the smooth cubic curve $D$ uniquely if and only if the $j$-invariant of $D$ is non-zero.
• The set of jumping lines of $\mathcal{T}(-\log D)$ coincides with the Cayleyan curve of $D$, which is a Hesse cubic in the dual plane.
• When $j(D) \neq 0$, the Cayleyan curve is smooth and determines $D$ up to three possibilities, which are resolved by the set of jumping cubic curves.
• When $j(D) = 0$, the Cayleyan curve degenerates into three lines in general position, and the sheaf $\mathcal{T}(-\log D)$ fails to be Torelli.
• A family of smooth cubics $az_0^3 + bz_1^3 + cz_2^3 = 0$ with $a,b,c \in \mathbb{C}^\times$ all have the same Cayleyan curve $\alpha_0\alpha_1\alpha_2 = 0$, hence isomorphic logarithmic sheaves, showing the Torelli property fails in this case.