Rank 3 quadratic generators of Veronese embeddings: the characteristic 3 case

Abstract

This paper investigates property QR(3) for Veronese embed-dings over an algebraically closed field of characteristic 3. We determine the rank index of (P-n, O-Pn (d)) for all n >= 2, d >= 3, proving that it equals 3 in these cases. Our approach adapts the inductive framework of [11], 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 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 of P-r of dimension at least 3, the rank index of (X, 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 char(K) not equal 2. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.