Let \(\mathrm{PG}(d,q)\) be a projective space of dimension \(d\) over the Galois field of order \(q\). Let \(\Sigma\) (resp. \(\tilde \Sigma\)) be a family of \(q^n+q^{n-1}+\ldots+q+1\) (resp. \(q^m+q^{m-1}+\ldots+q+1\)) subspaces of \(\mathrm{PG}(d,q)\) of dimension \(m\) (resp. \(n\)). The pair \((\Sigma, \tilde \Sigma)\) is a Segre pair with parameters \((q;m,n,d)\) if each member of \(\Sigma\) intersects each member of \(\tilde \Sigma\) in precisely one point, any two elements of \(\Sigma\) (resp. \(\tilde \Sigma\)) are disjoint and both families generate \(\mathrm{PG}(d,q)\).
Let \(p_1\) and \(p_2\) be points of \(\mathrm{PG}(n,q)\) and \(\mathrm{PG}(m,q)\) respectively. Let \(p_1*p_2\) be a point of \(\mathrm{PG}(nm+m+n,q)\) obtained multiplying the column associated with \(p_1\) with the row associated with \(p_2\). The Segre variety of the pair of the projective spaces \((\mathrm{PG}(n,q),\mathrm{PG}(m,q))\) is the subset of the points \(p_1*p_2 \in \mathrm{PG}(nm+n+m,q)\), with \(p_1 \in \mathrm{PG}(n,q)\), \(p_2 \in \mathrm{PG}(m,q)\). Any Segre variety contains two families of \(m\)-spaces and \(n\)-spaces forming a Segre pair.
The authors prove that also the converse is true, under the condition \(d\geq mn+m+n\).
In the paper a characterization of Segre pairs in terms of projection of some Segre varieties is given.
Finally, a third characterization of Segre varieties in terms of the \(3\)-spaces intersecting the given set in hyperbolic quadrics is proven. This last result holds also in the case of non-finite Segre varieties.