  In this paper we study finite dimensional algebras, in particular finite
semifields, through their correspondence with nonsingular threefold tensors. We
introduce a alternative embedding of the tensor product space into a projective
space. This model allows us to understand tensors and their contractions in a
new geometric way, relating the contraction of a tensor with a natural subspace
of a subgeometry. This leads us to new results on invariants and
classifications of tensors and algebras and on nonsingular fourfold tensors. A
detailed study of the geometry of this setup for the case of the threefold
tensor power of a vector space of dimension two over a finite field
surprisingly leads to a new construction of quasi-hermitian varieties in
$\mathrm{PG}(3,q^2)$.
