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The reciprocal complement of a curve

Dario Spirito

TL;DR

The paper investigates the reciprocal complement $\mathcal{R}(D)$ of a finitely generated one-dimensional $k$-algebra $D$, reframing it through the geometry of an affine curve $X$ with $k[X]\cong D$. It establishes that, for realizations regular at infinity, $\mathcal{R}(D)\neq\mathcal{Q}(D)$ if and only if the projective closure satisfies $|\overline{X}\setminus X|=1$, and connects the non-field case to the Weierstrass semigroup at infinity; in the Dedekind setting, $\mathcal{R}(D)$ is a DVR precisely when $\overline{X}$ has genus $0$. The results provide a concrete geometric interpretation of the guerrieri-recip criterion and relate valuation-theoretic properties of $\mathcal{R}(D)$ to classical invariants like the Weierstrass semigroup and genus, with illustrative examples from curves $x^n+y^n-1$ and elliptic curves. This bridges algebraic properties of $D$ with the projective geometry of its associated curve.

Abstract

We give a geometric interpretation of the reciprocal complement of an integral domain $D$ in the case $D$ is a one-dimensional finitely generated algebra over an algebraically closed field.

The reciprocal complement of a curve

TL;DR

The paper investigates the reciprocal complement of a finitely generated one-dimensional -algebra , reframing it through the geometry of an affine curve with . It establishes that, for realizations regular at infinity, if and only if the projective closure satisfies , and connects the non-field case to the Weierstrass semigroup at infinity; in the Dedekind setting, is a DVR precisely when has genus . The results provide a concrete geometric interpretation of the guerrieri-recip criterion and relate valuation-theoretic properties of to classical invariants like the Weierstrass semigroup and genus, with illustrative examples from curves and elliptic curves. This bridges algebraic properties of with the projective geometry of its associated curve.

Abstract

We give a geometric interpretation of the reciprocal complement of an integral domain in the case is a one-dimensional finitely generated algebra over an algebraically closed field.
Paper Structure (2 sections, 2 theorems)

This paper contains 2 sections, 2 theorems.

Table of Contents

  1. Introduction
  2. Results

Key Result

Theorem 2.1

Let $k$ be an algebraically closed field, and let $D$ be a finitely generated integral one-dimensional $k$-algebra. Then, the following are equivalent: Moreover, if these conditions hold and $\overline{X}\setminus X=\{p\}$, then the integral closure of $\mathcal{R}(D)$ is the local ring $\mathcal{O}_{\overline{X},p}$.

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Proposition 2.6
  • proof