Intersections of the Hermitian surface with irreducible quadrics in even characteristic. (English) Zbl 1368.51006
Let \(\mathrm{PG}(3,q^2)\) be the three dimensional projective space over the Galois field \(\mathrm{GF}(q^2)\). In this work, the authors determine the possible intersection sizes of a Hermitian surface \(\mathcal{H}\) with an irreducible quadric of \(\mathrm{PG}(3, q^2)\) sharing at least a tangent plane at a common nonsingular point when \(q\) is even, extending the arguments of [the authors, Finite Fields Appl. 30, 1–13 (2014; Zbl 1301.05050)] to the case of even characteristic. Due to the transitive action of \(\mathrm{PGU}(4, q)\) on \(\mathcal{X}\), the authors investigate the size of the following variety
\[
\begin{cases} z^{q} + z = x^{q+1} + y^{q+1}\\ z = ax^2 + by^2 + cxy + dx + ey + f,\end{cases}
\]
where \(a,b,c,d,e,f \in\mathrm{GF}(q^2)\). They mainly use algebraic methods, but in some cases geometric and combinatorial arguments are needed. These results have applications in determining the minimum weight of functional codes arising from Hermitian surfaces and quadrics in \(\mathrm{PG}(3, q^2)\).
Reviewer: Daniele Bartoli (Perugia)
MSC:
| 51E20 | Combinatorial structures in finite projective spaces |
Citations:
Zbl 1301.05050Software:
Book3264ExamplesReferences:
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