Ball, Simeon and Lavrauw, Michel (2020) Arcs and tensors. Designs, Codes, and Cryptography, 88 (1). pp. 1731. ISSN 09251022 (Print) 15737586 (Online)
This is the latest version of this item.
Official URL: http://dx.doi.org/10.1007/s1062301900668z
Abstract
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 k1), 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(Y1, ..., Yk1) (in k(k  1) variables) where Yj = (Xj1, ..., Xjk), which is homogeneous of degree t in each of the ktuples 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 SegreBlokhuisBruenThas hypersurface associated to an arc of hyperplanes in PG(k  1, q).
Item Type:  Article 

Uncontrolled Keywords:  Arcs; MDS codes 
Subjects:  Q Science > QA Mathematics > QA150272.5 Algebra Q Science > QA Mathematics > QA440 Geometry. Trigonometry. Topology 
Divisions:  Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences 
Depositing User:  Michel Lavrauw 
Date Deposited:  14 Sep 2020 12:30 
Last Modified:  25 Jul 2023 14:37 
URI:  https://research.sabanciuniv.edu/id/eprint/40071 
Available Versions of this Item

Arcs and tensors. (deposited 01 Aug 2019 22:57)
 Arcs and tensors. (deposited 14 Sep 2020 12:30) [Currently Displayed]