Segre embeddings and finite semifields

Each embedded product space PG(n,q)×PG(n,q) in an (n2+n−1)-dimensional projective space is obtained by projecting the Segre variety Sn,n,q from an n-subspace δ skew with its first secant variety (Zanella, 1996 [22]). On the other hand, when δ is skew with the (n−1)-th secant variety, it determines a semifield of order qn+1 whose center contains Fq (Lavrauw, 2011 [17]). A relationship arises between a particular class of embeddings of PG(n,q)×PG(n,q) in PG(n2+n−1,q) and semifields of the above type. For this reason, such embeddings will be called semifield embeddings. In this paper we establish this connection and prove that projectively equivalent semifield embeddings that do not exchange subspaces of different kind are related to isotopic semifields, and conversely. Exchanging the order in the product leads to the transition from a semifield to its transpose.