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Asymptotic Syzygies of Weighted Projective Spaces

Boyana Martinova

TL;DR

The paper extends the Ein–Erman–Lazarsfeld monomial syzygy framework to Veronese embeddings of the weighted projective space $\mathbb{P}(1^n,2)$, establishing explicit nonvanishing ranges for Betti entries in the asymptotic regime $d\gg0$. By refining the EEL method to accommodate nonstandard grading, using Artinian reduction and careful monomial tracking, it derives left and right endpoints $F_q(d)$ and $B_q(d)$ that bound where $\beta_{i,i+q}$ can be nonzero for odd and even $d$. It also proves that, in the odd case, the asymptotic density $\rho_q(M)$ of nonzero entries approaches 1 within the allowed rows, paralleling the original Ein–Lazarsfeld phenomenon for standard projective spaces. Regularity computations underpin the row-bounding strategy, showing how parity influences the number of Betti-table rows and enabling a row-overlap argument to ensure continuity of nonzero blocks. The work highlights both the tractability of $\mathbb{P}(1^n,2)$ and the complexities that arise for more general weighted spaces, outlining concrete challenges for future extensions.

Abstract

By adapting methods of Ein-Erman-Lazarsfeld, we prove an analogue of the Ein-Lazarsfeld result on asymptotic syzygies for Veronese embeddings, in the setting of weighted projective spaces of the form $\mathbb{P}(1^n,2)$.

Asymptotic Syzygies of Weighted Projective Spaces

TL;DR

The paper extends the Ein–Erman–Lazarsfeld monomial syzygy framework to Veronese embeddings of the weighted projective space , establishing explicit nonvanishing ranges for Betti entries in the asymptotic regime . By refining the EEL method to accommodate nonstandard grading, using Artinian reduction and careful monomial tracking, it derives left and right endpoints and that bound where can be nonzero for odd and even . It also proves that, in the odd case, the asymptotic density of nonzero entries approaches 1 within the allowed rows, paralleling the original Ein–Lazarsfeld phenomenon for standard projective spaces. Regularity computations underpin the row-bounding strategy, showing how parity influences the number of Betti-table rows and enabling a row-overlap argument to ensure continuity of nonzero blocks. The work highlights both the tractability of and the complexities that arise for more general weighted spaces, outlining concrete challenges for future extensions.

Abstract

By adapting methods of Ein-Erman-Lazarsfeld, we prove an analogue of the Ein-Lazarsfeld result on asymptotic syzygies for Veronese embeddings, in the setting of weighted projective spaces of the form .
Paper Structure (13 sections, 23 theorems, 94 equations, 1 table)

This paper contains 13 sections, 23 theorems, 94 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Ein-Erman-Lazarsfeld Monomial Syzygy Method
  3. Extending The EEL Method to Weighted Projective Spaces
  4. Regularity Computations for Veronese Embeddings of General Weighted Projective Spaces
  5. Notation
  6. Artinian Reduction
  7. Main Results
  8. Front of the Betti Table Results for Odd Veronese Degree
  9. Back of the Betti Table Results for Odd Veronese Degree
  10. Overlap for Odd Veronese Degree
  11. Results for Even Veronese Degrees
  12. Main Theorems
  13. Challenges for Other Weighted Projective Spaces

Key Result

Theorem 1

Consider the $d$th Veronese embedding $\varphi \colon \mathbb{P}(1^n,2) \xrightarrow[]{|\mathcal{O}(d)|} \operatorname{Proj}(S)$, where $S$ is some (possibly nonstandard graded) polynomial ring with $N+1$ variables. For each row $q$ of the Betti table, there exist constants $c_q$ and $C_q$ such that

Theorems & Definitions (59)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Conjecture 4: Erman-Martinova
  • Lemma 2.1
  • Example 2.2
  • Example 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 49 more