Arcs and tensors

To an arc A of PG ( k - 1 , q ) of size q + k - 1 - t we associate a tensor in ⟨ ν k , t ( A ) ⟩ ⊗ k - 1 , where ν k , t denotes the Veronese map of degree t defined on PG ( k - 1 , q ) . As a corollary we prove that for each arc A in PG ( k - 1 , q ) of size q + k - 1 - t , which is not contained in a hypersurface of degree t , there exists a polynomial F ( Y 1 , … , Y k - 1 ) (in k ( k - 1 ) variables) where Y j = ( X j 1 , … , X jk ) , which is homogeneous of degree t in each of the k -tuples of variables Y j , which upon evaluation at any ( k - 2 ) -subset S of the arc A gives a form of degree t on PG ( k - 1 , q ) whose zero locus is the tangent hypersurface of A at S , i.e. the union of the tangent hyperplanes of A at S . This generalises the equivalent result for planar arcs ( k = 3 ), proven in [ 2 ], to arcs in projective spaces of arbitrary dimension. A slightly weaker result is obtained for arcs in PG ( k - 1 , q ) of size q + k - 1 - t which are contained in a hypersurface of degree t . We also include a new proof of the Segre–Blokhuis–Bruen–Thas hypersurface associated to an arc of hyperplanes in PG ( k - 1 , q ) .