Arcs and tensors

Abstract

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, 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).

Arcs in finite projective spaces

“…, α t be t linear forms whose kernels are the t hyperplanes which meet A in precisely S, see Lemma 12. Define, up to a scalar factor, a homogeneous polynomial of degree t, f S (X) = t i=1 α i (X), (6) where X = (X 1 , . .…”