Holey Segre Varieties

Holey Segre varieties are introduced, which generalize classical Segre varieties and whose existence is suggested by the fact that any non-prime finite field contains proper subfields. More precisely, a holey Segre variety is the tensor product PG(n, q) ⊗ PG(m,qk ). An algebraic description of such an object is given and its structure is investigated. Furthermore, special choices of the parameters are considered, in which cases a holey Segre variety admits a partition into classical Segre varieties. Also, some holey Segre varieties are isomorphic to non-canonical subgeometries embedded in suitable Segre varieties.