Arcs and tensors

Abstract

To an arc A of PG(k-1,q) of size q+k-1-t we associate a tensor in ¿¿k,t(A)¿¿k-1 , where ¿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(Y1,…,Yk-1) (in k(k-1) variables) where Yj=(Xj1,…,Xjk) , which is homogeneous of degree t in each of the k-tuples of variables Yj , 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) .