Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weighted Veronese Rings via Convex Semigroups

Bek Chase, Luca Fiorindo, Thiago Holleben, Emanuela Marangone, Thai Thanh Nguyen, Alexandra Seceleanu, Srishti Singh

Abstract

We determine properties of two-dimensional normal affine semigroup rings, and in particular of weighted Veronese rings, including determinantal presentation, Gröbner basis, graded Hilbert series and graded Betti numbers, the structure of their associated graded rings, and their Koszul property. We give examples in higher dimensions illustrating that the first and last properties may fail. Our approach leverages convex monomial ideals as introduced in Herzog-Qureshi-Saem(2019), which give rise to convex semigroups.

Weighted Veronese Rings via Convex Semigroups

Abstract

We determine properties of two-dimensional normal affine semigroup rings, and in particular of weighted Veronese rings, including determinantal presentation, Gröbner basis, graded Hilbert series and graded Betti numbers, the structure of their associated graded rings, and their Koszul property. We give examples in higher dimensions illustrating that the first and last properties may fail. Our approach leverages convex monomial ideals as introduced in Herzog-Qureshi-Saem(2019), which give rise to convex semigroups.
Paper Structure (8 sections, 15 theorems, 68 equations, 1 figure)

This paper contains 8 sections, 15 theorems, 68 equations, 1 figure.

Table of Contents

  1. Background
  2. Convex ideals
  3. Convex semigroup rings
  4. Two-dimensional Veronese rings
  5. Two segment Veronese rings
  6. Higher-dimensional Veronese rings
  7. Failure of determinantal presentations
  8. Failure of the Koszul property

Key Result

Theorem 1

Let $S$ be a two-dimensional normal affine semigroup ring. Then If additionally $S$ is a convex semigroup ring (in particular if $S=V_{{\bf w},d}$) and $I=(x^{a_i}y^{b_i} : 1\leq i\leq s)$ is the ideal generated by the minimal monomial ${\mathbbm k}$-algebra generators of $S$ then

Figures (1)

  • Figure 1: Exponent vectors of the generators of $V_{(1,29),47}$ and corresponding matrix.

Theorems & Definitions (38)

  • Theorem
  • Definition 1.1
  • Theorem 1.2: ConvexIdeals
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Example 1.5
  • Definition 2.1
  • Example 2.2
  • ...and 28 more