Saerom Sim

DGIST, Daegu, Korea

On the quadratic rank of Veronese embeddings in all characteristics

As a special case of symmetric tensor rank, the quadratic rank of the defining ideals for classical varieties has recently been studied. Especially, since determinantally presented varieties automatically satisfy property $QR(4)$ (i.e., their ideals are generated by quadrics of rank at most 4), it is natural to ask whether they further satisfy $QR(3)$. With the help of the $Q$-morphism introduced by Han-Lee-Moon-Park, we can now better understand this property. In this work, we complete the classification of quadratic rank for Veronese embeddings by resolving the remaining cases in characteristic 3. Furthermore, we demonstrate how the geometric structure of the rank 3 locus determines whether the variety satisfies $QR(3)$ or fails.