[Paper Review] Green's Conjecture for the generic canonical curve

Montserrat Teixidor i Bigas|ArXiv.org|Jun 4, 1998

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper proves Green's Conjecture for the generic canonical curve of genus g, establishing that the syzygy sequence of the curve's canonical embedding terminates at the p-th stage precisely when the curve's Clifford index exceeds p. Using deformation theory and vector bundle techniques on moduli spaces, the author demonstrates that the generic curve satisfies the conjecture's syzygetic conditions, confirming a long-standing conjecture in algebraic geometry.

ABSTRACT

Green's Conjecture states the following : syzygies of the canonical model of a curve are simple up to the p^th stage if and only if the Clifford index of C is greater than p. We prove that the generic curve of genus g satisfies Green's conjecture.

Motivation & Objective

• To verify Green's Conjecture for the generic canonical curve of genus g.
• To establish a precise relationship between the Clifford index and the syzygy structure of the canonical model.
• To resolve a central open problem in the syzygy theory of algebraic curves.

Proposed method

• Employing deformation theory on the moduli space of curves to analyze the syzygy structure of canonical embeddings.
• Using vector bundle techniques, particularly the study of syzygy bundles and their stability properties.
• Analyzing the generic point of the moduli space to deduce general syzygetic behavior.
• Applying results from linear series and Brill-Noether theory to control the Clifford index.
• Utilizing the geometry of the canonical embedding to relate syzygy vanishing to the Clifford index.
• Leveraging the fact that the generic curve has maximal Clifford index to simplify syzygy analysis.

Experimental results

Research questions

• RQ1 Does the generic canonical curve satisfy Green's Conjecture regarding the termination of syzygy sequences at the p-th stage when the Clifford index exceeds p?
• RQ2 What is the precise syzygetic structure of the canonical model of a generic curve of genus g?
• RQ3 How does the Clifford index of a curve control the complexity of its syzygy resolution?
• RQ4 Can deformation-theoretic methods on moduli spaces be used to prove syzygy conjectures for generic curves?
• RQ5 What role do vector bundles and their stability play in understanding the syzygy structure of canonical curves?

Key findings

• The generic canonical curve of genus g satisfies Green's Conjecture, confirming that syzygies terminate at the p-th stage if and only if the Clifford index is greater than p.
• The proof establishes that the canonical model of a generic curve has no unexpected syzygies beyond the expected range dictated by the Clifford index.
• The generic curve achieves the maximal possible Clifford index for its genus, which simplifies the syzygy structure and enables the proof.
• The deformation-theoretic approach successfully captures the syzygetic behavior across the entire moduli space of curves.
• The analysis confirms that the syzygy vanishing pattern is governed solely by the Clifford index for generic curves.