Rank 3 Quadratic Generators of Veronese Embeddings: The Characteristic 3 Case
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 ${\rm char}(\mathbb{K}) \ne 2$.