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Section Rings of $\mathbb{Q}$-Divisors on Genus $1$ Curves

Michael Cerchia, Jesse Franklin, Evan O'Dorney

Abstract

We compute generators and relations for the section ring of a rational divisor on an elliptic curve. Our technique generalizes the work of O'Dorney (in genus zero) and Voight--Zureick-Brown (for specific divisors arising from the study of stacky curves). For effective divisors supported on at most two points, we give explicit descriptions of the generators and the leading terms of the relations for a minimal presentation. As in the genus zero case, the generators are parametrized by best lower approximations to the coefficients, but there are added wrinkles. Following Landesman, Ruhm and Zhang we can bound the degrees of generators for the section ring of an effective divisor supported at any finite number of points.

Section Rings of $\mathbb{Q}$-Divisors on Genus $1$ Curves

Abstract

We compute generators and relations for the section ring of a rational divisor on an elliptic curve. Our technique generalizes the work of O'Dorney (in genus zero) and Voight--Zureick-Brown (for specific divisors arising from the study of stacky curves). For effective divisors supported on at most two points, we give explicit descriptions of the generators and the leading terms of the relations for a minimal presentation. As in the genus zero case, the generators are parametrized by best lower approximations to the coefficients, but there are added wrinkles. Following Landesman, Ruhm and Zhang we can bound the degrees of generators for the section ring of an effective divisor supported at any finite number of points.
Paper Structure (14 sections, 15 theorems, 91 equations, 5 figures, 1 table)

This paper contains 14 sections, 15 theorems, 91 equations, 5 figures, 1 table.

Table of Contents

  1. Motivation
  2. Main results
  3. Acknowledgements
  4. The one-point case
  5. Generators
  6. Relations
  7. The effective two-point case
  8. Generators
  9. Relations
  10. The unequal ceilings case
  11. The equal ceilings case
  12. The subtle behavior of the ineffective two-point case
  13. A conjecture on generators
  14. Arbitrary effective Q-divisors

Key Result

Theorem 1.1

Let $D = \alpha (\infty)$ be a $\mathbb{Q}$-divisor supported on one point of $\mathbb{P}^1$. Denote by $t$ a coordinate on $\mathbb{P}^1$ having a simple pole at this point. Let be the nonnegative best lower approximations to $\alpha$. Then $S_D$ has a minimal presentation consisting of the $r+1$ generators $f_i = t^{c_i}u^{d_i}$ and $\binom{r}{2}$ relations of the form for each $(i,j)$ with $j

Figures (5)

  • Figure 1: The section ring of $D = (P)$, which has three generators
  • Figure 2: Cases covered by Lemma \ref{['lem:unique']}, where a generator of $M$ has a point of $M$ directly above it. The bullets indicate generators, annotated with their type (items a--d of Theorem \ref{['thm:1point_gens']}).
  • Figure 3: Example bases for $S_D$ with $D$ as in Lemma \ref{['lemma: Z-basis for ineffective 2pt case']}
  • Figure 4: Generators for $S_D$ labeled according to Conjecture \ref{['conj:2point ineffective']} when $D=\frac{13}{5}P^{(1)}-\frac{1}{7}P^{(2)}.$
  • Figure 5: Generators for $S_D$ labeled according to Conjecture \ref{['conj:2point ineffective']} where $D=\frac{2}{3}P^{(1)}-\frac{3}{5}P^{(2)}.$

Theorems & Definitions (45)

  • Theorem 1.1: O'Dorney, Theorem 4
  • Theorem 1.2: see Theorems \ref{['thm:1point_gens']}, \ref{['thm:1point_rels']}
  • Theorem 1.3: see Theorems \ref{['thm:2point_gens']}, \ref{['thm:2point_rels_unequal']}, \ref{['thm:2point_rels_equal']}
  • Theorem 1.4
  • Example 2.1
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof : Proof of Lemma \ref{['lem:irred_M']}
  • ...and 35 more