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Syzygies of canonical ribbons on higher genus curves

Anand Deopurkar, Jayan Mukherjee

TL;DR

This work analyzes the syzygies of canonical embeddings of ribbons on curves, linking geometric Clifford indices (LCliff, RCliff) to algebraic Green-type invariants via Koszul cohomology. It proves that for general ribbons with large arithmetic genus $p_a$ on a general curve, $LCliff = gon - 2 = RCliff = \left\lfloor\frac{p_a-1}{2}\right\rfloor$, and derives Green's conjecture for these ribbons, with consequences for double covers. It develops blow-up and gonality stratifications, connects these to secant varieties and to stability of the conormal bundle, and reduces key vanishing questions to vanishings in auxiliary modules $M^p$, offering a coherent framework for both general and split ribbons. The paper also provides detailed bounds and exact values for split ribbons, shows that equality of Clifford indices in the split case implies Green's conjecture for associated double covers, and relates canonical ribbons to deformations of $K3$ carpets, enriching the landscape of higher-genus ribbon geometry and its syzygies.

Abstract

We study the syzygies of the canonical embedding of a ribbon $\widetilde{C}$ on a curve $C$ of genus $g \geq 1$. We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic genus $p_a$ on a general curve of genus $g$ with $p_{a} \geq \operatorname{max}\{3g+7, 6g-4\}$. Among non-general ribbons, the case of split ribbons is particularly interesting. Equality of the two Clifford indices for a split ribbon is related to the gonality conjecture for $C$ and it implies Green's conjecture for all double covers $C'$ of $C$ with $g(C') \geq \textrm{max}\{3g+2, 6g-4\}$. We reduce it to the vanishing of certain Koszul cohomology groups of an auxiliary module of syzygies associated to $C$, which may be of independent interest.

Syzygies of canonical ribbons on higher genus curves

TL;DR

This work analyzes the syzygies of canonical embeddings of ribbons on curves, linking geometric Clifford indices (LCliff, RCliff) to algebraic Green-type invariants via Koszul cohomology. It proves that for general ribbons with large arithmetic genus on a general curve, , and derives Green's conjecture for these ribbons, with consequences for double covers. It develops blow-up and gonality stratifications, connects these to secant varieties and to stability of the conormal bundle, and reduces key vanishing questions to vanishings in auxiliary modules , offering a coherent framework for both general and split ribbons. The paper also provides detailed bounds and exact values for split ribbons, shows that equality of Clifford indices in the split case implies Green's conjecture for associated double covers, and relates canonical ribbons to deformations of carpets, enriching the landscape of higher-genus ribbon geometry and its syzygies.

Abstract

We study the syzygies of the canonical embedding of a ribbon on a curve of genus . We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic genus on a general curve of genus with . Among non-general ribbons, the case of split ribbons is particularly interesting. Equality of the two Clifford indices for a split ribbon is related to the gonality conjecture for and it implies Green's conjecture for all double covers of with . We reduce it to the vanishing of certain Koszul cohomology groups of an auxiliary module of syzygies associated to , which may be of independent interest.

Paper Structure

This paper contains 16 sections, 34 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Organization and conventions
  4. Canonical embedding of ribbons on curves of higher genus
  5. Linear Series Clifford index and resolution Clifford index of a ribbon
  6. Semicontinuity of gonality and Clifford index
  7. Green-Lazarsfeld non-vanishing theorem for ribbons
  8. Blow-up index stratification of the space of ribbons
  9. Blow-up index of a ribbon as a pushout and relations to secant variety
  10. Blow-up index as pullback and relations to stability of $\Omega_{\widetilde{C}|_C}$
  11. Blow-up index, gonality, and linear series Clifford index
  12. Gonality stratification of the space of ribbons
  13. Green's conjecture and resolution Clifford index for a general ribbon on a general curve
  14. Gonality and Clifford index of a split ribbon
  15. Linear series Clifford index of a split ribbon
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $C$ be a general curve of genus $g$. Fix a line bundle $L$ on $C$. Let $\widetilde{C}$ be a general ribbon with conormal bundle $L$. Let $p_{a}(\widetilde{C})$ be the arithmetic genus, $\mathop{\mathrm{gon}}\nolimits(\widetilde{C})$ the gonality, $\mathop{\mathrm{LCliff}}\nolimits(\widetilde{C})

Theorems & Definitions (72)

  • Definition 1.1
  • Theorem 1.2: \ref{['odd genus maximum blow-up index']}, \ref{['generic green even']}, \ref{['any genus and any blowup index']} in the main text
  • Theorem 1.3: \ref{['gonality stratification']} and \ref{['inclusion of gonality stratification in secant variety']}
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • ...and 62 more