Green's conjecture

Green's conjecture predicts that the Clifford index of a smooth projective curve equals the first index where a Koszul cohomology group of its canonical embedding is nonzero.

Connect

Koszul cohomology: A family of cohomology groups measuring algebraic relations among generators of a graded module. Its groups encode the syzygies whose vanishing Green's conjecture predicts.
Algebraic curve: A one-dimensional algebraic variety, often studied as a smooth projective curve. The conjecture is formulated for smooth projective curves.
Mark Green (mathematician): An American mathematician known for work in complex analysis, algebraic geometry, and syzygies. He formulated the conjecture connecting canonical syzygies with the Clifford index.
Minimal free resolution: A free resolution whose differential maps have entries in the homogeneous maximal ideal. Koszul cohomology records the graded pieces of the canonical coordinate ring's minimal resolution.
Gonality: The least degree of a nonconstant morphism from a curve to the projective line. Gonality measures one type of special linear series, while the Clifford index captures a broader range.
Canonical embedding: The map of a curve to projective space defined by its complete canonical linear system. The conjecture concerns the syzygies of this embedding.
Genus (mathematics): A nonnegative integer measuring the complexity of a compact Riemann surface or smooth projective curve. The genus determines the ambient projective space of the canonical embedding.
Green's linear syzygy theorem: A theorem describing linear syzygies of projective embeddings through Koszul cohomology. The conjecture emerged from Green's broader study of linear syzygies.
Canonical ring: The graded ring formed from all tensor powers of a curve's canonical line bundle. Its generators and relations encode the canonical embedding's syzygies.
Clifford's theorem: A bound on the number of sections of a special line bundle on a smooth curve. It constrains the special divisors summarized by the conjecture's Clifford index.