Rank 3 Quadratic Generators of Veronese Embeddings: The Characteristic 3 Case

TL;DR

This work addresses the rank-index problem for Veronese embeddings in characteristic $3$, proving that $\text{rank-index}((\mathbb P^n,\mathcal O_{{\mathbb P^n}}(d)))=3$ for all $n\ge2$ and $d\ge3$, and clarifying exceptional behavior of the second Veronese via a codimension ${n+1\choose 4}$ for the span of rank-$3$ quadrics. The authors adapt the HLMP inductive framework to char $3$, re-proving key lemmas and establishing that the ideal of the Veronese variety is generated by rank-$3$ quadrics in the appropriate range, with generation controlled by a finite set $\Gamma(n,d)$. They also analyze the second Veronese embedding to derive a precise codimension count, and they prove that for general complete intersections of quadrics of dimension at least $3$, the rank-index rises to $4$, demonstrating optimality and completing the classification for ${\rm char}(\mathbb K)\neq2$. Collectively, the results explain the peculiarities of QR$(3)$ in characteristic $3$ and furnish sharp bounds on rank-3 generation for Veronese varieties and related complete intersections.

Abstract

This paper investigates property QR(3) for Veronese embeddings over an algebraically closed field of characteristic $3$. We determine the rank index of $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (d))$ for all $n \geq 2$, $d \geq 3$, proving that it equals $3$ in these cases. Our approach adapts the inductive framework of [HLMP 2021], re-proving key lemmas for characteristic $3$ to establish quadratic generation by rank $3$ forms. We further compute the codimension of the span of rank $3$ quadrics in the space of quadratic equations of the second Veronese embedding, showing it grows as ${n+1 \choose 4}$. This provides a clear explanation of the exceptional behavior exhibited by the second Veronese embedding in characteristic $3$. Additionally, we show that for a general complete intersection of quadrics $X \subset \mathbb{P}^r$ of dimension at least $3$, the rank index of $(X,\mathcal{O}_X (2))$ is $4$, thereby confirming the optimality of our main bound. These results complete the classification of the rank index for Veronese embeddings when ${\rm char}(\mathbb{K}) \ne 2$.

Figures (1)

• Figure 1: Structure of the proof of Theorem 4.2 in HLMP

Theorems & Definitions (14)

• Theorem 1.1: Theorem 1.1 and Theorem 1.3 in HLMP, Theorem 1.1 in P22
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.3
• proof
• Proposition 3.1
• proof
• Theorem 3.2
• Lemma 3.4