The \(\psi\)-associate \(X^{\#}\) of a flat \(X\) in \(PG(n,2)\)
If \(\Psi\) is a hypersurface of \(PG(n,2),\) with equation \(Q(x)=0,\) where \(Q\) is a polynomial of reduced degree \(d>1,\) denote by \(T_Q\) the complete polarized form of \(Q.\) The \(\Psi-\)associate \(X^\sharp\) of an \(r-\)flat \(X\) is the set \[ X^{\sharp}=\{y\in PG(n,2)\mid T_Q(x_1,x_2,\dots,x_{d-1},y)=0 \;\forall x_1,x_2,\dots,x_{d-1} \in X\}. \] Then \(X^{\sharp}\) is a flat. For all \(r<d-2,\) we have \(X^{\sharp}=PG(n,2).\) If \(r=d-2,\) then \(X\subset X^{\sharp}.\) A \((d-2)-\) flat \(W\) is \textit{singular} if \(W^{\sharp}=PG(n,2).\) Several choices of \(\Psi\) are visited, many of them are cubic hypersurfaces in \(PG(n,2).\) Let \(S_{1,2,2}\) be the Segre variety of \(PG(5,2)\) and let \(\mathcal R\) be the set of all planes contained in \(S_{1,2,2}.\) The \(48\) planes external to \(S_{1,2,2}\) fall into eight pairs of ordered triplets \(\{(P_1,R_1,S_1),(P_2,R_2,S_2)\}\) such that \(PG(5,2)\setminus \Psi=P_1\cup R_1\cup S_1\cup P_2\cup R_2\cup S_2\) and \(P_i^{\sharp}=R_i,\) \(R_i^{\sharp}=S_i,\) \(S_i^{\sharp}=P_i,\) \(i=1,2.\) Further the lines \(L\) of \(PG(5,2),\) which are singular, form a line-spread of \(PG(5,2)\). In \(PG(5,2),\) up to projective equivalence, there is just one kind of maximal partial spread, which has size \(5\). If \(\Sigma_5\) is a such partial spread and \(\Psi _5=\bigcup_{X \in \Sigma_5}X,\) then \(\Psi_5\) is a hypersurface of \(PG(5,2)\) and it is shown that one plane \(W\) of \(\Sigma_5\) is singled out by the properties that every line \(L\) of \(W\) if singular.
scientific article; zbMATH DE number 5536855