Multigraded Betti numbers of Veronese embeddings

TL;DR

We address the problem of describing the multigraded Betti numbers $\beta_{p,\mathbf{b}}$ of the $d$-uple Veronese embedding of $\mathbb{P}^m$ by translating to the reduced homology of the simplicial complexes $\Delta_{\mathbf{b}}$ via Hochster's formula and analyzing them with Forman's discrete Morse theory. It proves vanishing theorems: if $|\mathbf{b}| = dj$ with $\mathbf{b}_0 \ge A_j$ or with $j \ge d+1$ and $\mathbf{b}_0 \le \tilde{l}_j$, then all Betti numbers vanish. It also obtains non-vanishing patterns on explicit regions, computes exact Betti numbers in boundary cases (notably when $\mathbf{b}_0 = A_{p+1}-1$), and extends the framework to higher-dimensional projective spaces, establishing the optimality of the bounds $A_{p+1}$. The results illuminate the syzygies of Veronese embeddings and provide sharp combinatorial criteria for the presence of nontrivial multigraded syzygies.

Abstract

In this paper, we study the multigraded Betti numbers of Veronese embeddings of projective spaces. Due to Hochster's formula, we interpret these multigraded Betti numbers in terms of the homology of certain simplicial complexes. By analyzing these simplicial complexes and applying Forman's discrete Morse theory, we derive vanishing and non-vanishing results for these multigraded Betti numbers.

Figures (1)

• Figure 1: Multigraded Betti numbers $\beta_{4,\mathbf{b}}$ for $|\mathbf{b}|=15$.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Theorem 2.3: Hochster's formula
• Definition 2.4: zbMATH02041483
• Theorem 2.5: zbMATH02041483
• Example 2.6
• Theorem 2.7: zbMATH02041483