Green's conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $C$ be a smooth connected projective non-hyperelliptic curve over $\mathbb C$ of genus $g\geq3$, and let $K_C$ be its canonical line bundle. Define \[\operatorname{Cliff}(C)=\min_{\substack{L\in\operatorname{Pic}(C)\\h^0(C,L)\geq2,\ h^1(C,L)\geq2}}\left(\deg L-2(h^0(C,L)-1)\right). \] For genus three use the standard value $\operatorname{Cliff}(C)=1$ when the displayed minimization set is empty. Put $V=H^0(C,K_C)$ and let $K_{p,1}(C,K_C)$ be the middle cohomology of the Koszul complex \[\bigwedge^{p+1}V\otimes H^0(C,\mathcal O_C)\longrightarrow\bigwedge^pV\otimes H^0(C,K_C)\longrightarrow\bigwedge^{p-1}V\otimes H^0(C,K_C^{\otimes2}), \] whose differential alternately removes a vector and multiplies by its section; a negative exterior power is zero.

Statement recorded in the supplied source.

Does every such curve satisfy \[K_{p,1}(C,K_C)=0\quad\Longleftrightarrow\quad p\geq g-\operatorname{Cliff}(C)-1,\qquad p\geq1? \] Thus the length of the linear syzygy strand determines the Clifford index. [S2]

Short English statement

Does a curve’s Clifford index coincide with the first nonlinear syzygy of its canonical embedding?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1600011 (AMR-015-0011). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 G. Farkas, Progress on syzygies of algebraic curves, arXiv:1703.08056v2 (2017), §2 and §4, Definition 4.2 and Conjecture 4.4. https://arxiv.org/abs/1703.08056.