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The Prym-canonical Clifford index

Margherita Lelli-Chiesa, Martina Miseri

TL;DR

This work introduces and studies the Prym-canonical Clifford index $\mathrm{Cliff}_{\eta}(C)$ and its dimension for Prym curves $(C,\eta)$, establishing their basic nonnegativity, sharp upper bound $\left\lfloor\frac{g-1}{2}\right\rfloor$, and a Prym-Clifford-type classification. It proves a relation between $\mathrm{Cliff}_{\eta}$ and the classical Clifford index, and analyzes the $\iota$-invariant Clifford index of the étale double cover $\tilde{C}$, linking Prym data to the gonality of $C$. The paper then classifies Prym curves with low Prym-canonical Clifford index (0,1,2), including a complete account for bielliptic and hyperelliptic cases, and derives precise results for the Prym-canonical dimension in these regimes. For hyperelliptic Prym curves, a Verra-type description of $2$-torsion shows $\mathrm{Cliff}_{\eta}(C)=k-1$ with dimension $(0,0)$, while in general the Prym-canonical Clifford index attains its maximum on a general Prym curve, reflecting a strong link between Prym geometry and syzygies of Prym-canonical models. The findings illuminate how Prym data governs both linear series and the embedding properties of Prym-canonical curves, with implications for understanding Prym-related moduli and syzygetic behavior.

Abstract

We introduce two new invariants of Prym curves, the Prym-canonical Clifford index and the Prym-canonical Clifford dimension. The former is a nonnegative integer (according to Prym-Clifford's theorem), while the latter is a pair of nonnegative ordered integers. We classify Prym curves with Prym-canonical Clifford index equal to 0,1,2. By specialization to hyperelliptic curves, we compute the Prym-canonical Clifford index of a general Prym curve and show that its Prym-canonical Clifford dimension is (0,0).

The Prym-canonical Clifford index

TL;DR

This work introduces and studies the Prym-canonical Clifford index and its dimension for Prym curves , establishing their basic nonnegativity, sharp upper bound , and a Prym-Clifford-type classification. It proves a relation between and the classical Clifford index, and analyzes the -invariant Clifford index of the étale double cover , linking Prym data to the gonality of . The paper then classifies Prym curves with low Prym-canonical Clifford index (0,1,2), including a complete account for bielliptic and hyperelliptic cases, and derives precise results for the Prym-canonical dimension in these regimes. For hyperelliptic Prym curves, a Verra-type description of -torsion shows with dimension , while in general the Prym-canonical Clifford index attains its maximum on a general Prym curve, reflecting a strong link between Prym geometry and syzygies of Prym-canonical models. The findings illuminate how Prym data governs both linear series and the embedding properties of Prym-canonical curves, with implications for understanding Prym-related moduli and syzygetic behavior.

Abstract

We introduce two new invariants of Prym curves, the Prym-canonical Clifford index and the Prym-canonical Clifford dimension. The former is a nonnegative integer (according to Prym-Clifford's theorem), while the latter is a pair of nonnegative ordered integers. We classify Prym curves with Prym-canonical Clifford index equal to 0,1,2. By specialization to hyperelliptic curves, we compute the Prym-canonical Clifford index of a general Prym curve and show that its Prym-canonical Clifford dimension is (0,0).
Paper Structure (14 sections, 29 theorems, 79 equations)

This paper contains 14 sections, 29 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Preliminaries
  4. Definitions and first results
  5. The Prym-canonical Clifford index and dimension
  6. Prym-Clifford's theorem
  7. The étale double cover and its invariant Clifford index
  8. Classification of curves with low Prym-canonical Clifford index
  9. Background information on curves of low gonality
  10. Prym-canonical Clifford dimension of bielliptic curves
  11. Classification theorems
  12. Prym-Clifford index of a general Prym curve
  13. Prym-Clifford index of hyperelliptic Prym curves
  14. The case of general curves

Key Result

Theorem 1.1

(Prym--Clifford's theorem) Let $(C,\eta)$ be a Prym curve of genus $g\ge 2$. Then and equality holds if and only if $\omega_C\otimes \eta$ has some base points, that is, when $C$ is hyperelliptic and $\eta=\mathcal{O}_{C}(p-q)$ with $p$ and $q$ Weierstrass points.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Corollary 2.2
  • Remark 1
  • Remark 2
  • Theorem 2.3
  • ...and 57 more