On Severi varieties as intersections of a minimum number of quadrics

Let \({\mathscr{V}}\) be a variety related to the second row of Freudenthal-Tits Magic square in \(N\)-dimensional projective space over an arbitrary field. We show that there exist \(M\leq N\) quadrics intersecting precisely if and only exists subspace dimension \(N-M\) secant disjoint from Severi variety. present some examples such subspaces relatively large dimension. In particular, real numbers we Cartan (related exceptional group \({E_6}\)\((\mathbb R)\)) is set-theoretic intersection 15 quadrics.