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Smooth Herzog projective curves

Hang Huang, Yevgeniya Tarasova, Matteo Varbaro, Emily Witt

TL;DR

The paper proves that any connected, smooth Herzog projective curve over a field has genus zero, by linking the algebraic structure of Herzog ideals to combinatorial properties of associated graphs via Gröbner degenerations. It develops an inductive framework based on a graph invariant $ ext{ell}( riangle)$ and a Gröbner basis built from minimal non-faces, and shows that non-tree graphs force singularities in any Gröbner deformation. Consequently, a smooth connected Herzog curve forces the underlying graph to be a tree, aligning with the classical rational-normal curve in projective space. The work also clarifies that Gröbner smoothability is stricter than general smoothability and can depend on coordinates, offering insight into the interplay between initial ideals and geometric properties.

Abstract

We prove that smooth projective curves admitting a squarefree Groebner degeneration have genus 0.

Smooth Herzog projective curves

TL;DR

The paper proves that any connected, smooth Herzog projective curve over a field has genus zero, by linking the algebraic structure of Herzog ideals to combinatorial properties of associated graphs via Gröbner degenerations. It develops an inductive framework based on a graph invariant and a Gröbner basis built from minimal non-faces, and shows that non-tree graphs force singularities in any Gröbner deformation. Consequently, a smooth connected Herzog curve forces the underlying graph to be a tree, aligning with the classical rational-normal curve in projective space. The work also clarifies that Gröbner smoothability is stricter than general smoothability and can depend on coordinates, offering insight into the interplay between initial ideals and geometric properties.

Abstract

We prove that smooth projective curves admitting a squarefree Groebner degeneration have genus 0.

Paper Structure

This paper contains 3 sections, 5 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminary results
  3. Proof of the main result

Key Result

Proposition 2.1

With the above notation, if the Herzog projective variety $X\subseteq \mathbb{P}^n$ is smooth over $K$, then the following are equivalent:

Theorems & Definitions (16)

  • Conjecture 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Conjecture 2.2
  • Lemma 2.3
  • proof
  • proof
  • Remark 2.5
  • Remark 2.6
  • ...and 6 more