The geproci property in positive characteristic (Q6566375)

The author constructs new examples of geproci sets in positive characteristic. Let \(K\) be an algebraically closed field. A finite set of reduced points \(Z\) in \(\mathbb{P}^{n}_{K}\) is called \textit{geproci} if the projection \(\overline{Z}\) of \(Z\) from a general point \(P \in \mathbb{P}^{n}_{K}\) to a hyperplane \(H\) is a complete intersection in \(H\). A set \(Z \subset \mathbb{P}^{3}_{K}\) is a \(\{ \mu, \lambda\}\)-half grid if \(Z\) is a non-trivial \(\{\mu, \lambda\}\)-geproci set, i.e., the image of \(Z\) under a general projection into \(\mathbb{P}^{2}_{K}\) is the complete intersection of a curve of degree \(\mu\) and a curve of degree \(\lambda\), that is contained in a set of \(\mu\) mutually-skew lines which the property that each line contains \(\lambda\) points of \(Z\). It is worth noticing that the general projection of an \(\{\mu, \lambda\}\)-half grid is a complete intersection of a union of \(\mu\) lines and a degree \(\lambda\) curve which is not a union of lines. The author, in order to construct new examples of geproci sets in positive characteristic, decided to use spreads. More precisely, let \(S\) be a set of \((t-1)\)-dimensional linear subspaces of \(\mathbb{P}^{2t-1}_{k}\) each of which is defined over (any field) \(k\). We call \(S\) as a spread if each point of \(\mathbb{P}^{2t-1}_{k}\) is contained in one and only one member of \(S\). Over a finite field, spreads always exist for each \(t\geq 1\). In the case \(t=2\), a spread in \(\mathbb{P}^{3}_{k}\) will be a set of mutually-skew lines defined over \(k\) that cover \(\mathbb{P}^{3}_{k}\).\N\NTheorem A. Let \(\mathbb{F}_{q}\) be the field of size \(q\) with \(q\) being some power of a prime number. Then \(Z = \mathbb{P}^{3}_{\mathbb{F}_{q}} \subset \mathbb{P}^{3}_{\overline{\mathbb{F}}_{q}}\) is a \((q+1, q^{2}+1)\)-geproci half grid.\N\N\noindent A partial spread of \(\mathbb{P}^{3}_{\mathbb{F}_{q}}\) with deficiency \(d\) is a set of \(q^{2}+1-d\) mutually-skew lines of \(\mathbb{P}^{3}_{\mathbb{F}_{q}}\). A maximal partial spread is a partial spread of positive deficiency that is not contained in any larger partial spread.\N\NTheorem B. The complement \(\widetilde{Z} \subset \mathbb{P}^{3}_{\mathbb{F}_{q}}\) of a maximal partial spread of deficiency \(d\) is a non-trivial \(\{q+1, d\}\)-geproci set. Furthermore, when \(d>q+1\), \(\widetilde{Z}\) is also not a half grid.