Arcs and tensors
Ball, Simeon and Lavrauw, Michel (2020) Arcs and tensors. Designs, Codes, and Cryptography, 88 (1). pp. 17-31. ISSN 0925-1022 (Print) 1573-7586 (Online)
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Official URL: http://dx.doi.org/10.1007/s10623-019-00668-z
To an arc A of PG(k - 1, q) of size q + k - 1 - t we associate a tensor in <nu(k,t) (A)>(circle times k-1), where nu(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, ..., Yk-1) (in k(k - 1) variables) where Y-j = (X-j1, ..., 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 , 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).
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