Two distinct Desarguesian (h-1)-spreads in PG(kh-1,q) sharing a pseudo-arc of size k+1 intersect precisely in the system of (h-1)-subspaces of a generalized Segre variety S^r_{kr-1,h-1}(q) for some proper divisor r of h.
Harris.Algebraic geometry
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Segre Varieties and Desarguesian Spreads
Two distinct Desarguesian (h-1)-spreads in PG(kh-1,q) sharing a pseudo-arc of size k+1 intersect precisely in the system of (h-1)-subspaces of a generalized Segre variety S^r_{kr-1,h-1}(q) for some proper divisor r of h.