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About how large are algebraic Betti numbers?

Daniel Erman

TL;DR

This work addresses the quantitative behavior of algebraic Betti numbers under very positive embeddings. It develops order-of-magnitude bounds for $\beta_i(M)$ by combining Boij–Söderberg decompositions with analytic control of pure diagrams, yielding universal lower and upper bounds in terms of codimension, projective dimension, and regularity. For Veronese embeddings of projective spaces and more generally for very ample line bundles, the bounds translate into explicit digit-count estimates, illustrating that Betti numbers can be enormous while their growth remains predictable in order of magnitude. The results highlight the power of Boij–Söderberg theory to illuminate asymptotic syzygy questions and connect homological invariants to combinatorial binomial structures, with concrete examples demonstrating the scale of the numbers involved.

Abstract

We use Boij-Söderberg theory to provide some order of magnitude bounds on algebraic Betti numbers.

About how large are algebraic Betti numbers?

TL;DR

This work addresses the quantitative behavior of algebraic Betti numbers under very positive embeddings. It develops order-of-magnitude bounds for by combining Boij–Söderberg decompositions with analytic control of pure diagrams, yielding universal lower and upper bounds in terms of codimension, projective dimension, and regularity. For Veronese embeddings of projective spaces and more generally for very ample line bundles, the bounds translate into explicit digit-count estimates, illustrating that Betti numbers can be enormous while their growth remains predictable in order of magnitude. The results highlight the power of Boij–Söderberg theory to illuminate asymptotic syzygy questions and connect homological invariants to combinatorial binomial structures, with concrete examples demonstrating the scale of the numbers involved.

Abstract

We use Boij-Söderberg theory to provide some order of magnitude bounds on algebraic Betti numbers.

Paper Structure

This paper contains 4 sections, 8 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on Boij-Söderberg Theory
  3. Proofs of Theorems
  4. Examples

Key Result

Theorem 1.1

Fix $n,d\geq 1$ and let $N=\binom{n+d}{d} - n-1$. For any $i$ we have

Figures (1)

  • Figure 1: The heuristic behind our results is that the numerical properties of a pure diagram shaped like the bottom figure (300 columns and 3 rows) will be approximately the same as for the top figure (300 columns and 1 row).

Theorems & Definitions (20)

  • Theorem 1.1
  • Example 1.2
  • Theorem 1.3
  • Example 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Example 2.2
  • Lemma 3.1
  • Example 3.2
  • Example 3.3
  • ...and 10 more