A Bertini type theorem for pencils over finite fields

Smoothness in Pencils of Hypersurfaces Over Finite Fields

“…For δ = 2, the conclusion is still true because 1 4 δ 2 = 1 and q ≥ 2. In particular, there exists an F q -line L which does not pass through any F q -point of X, as desired.…”

“…It is known that X is geometrically irreducible [4, Proposition 7.1] and the degree of X is by [4, Proposition 7.4]. Note that the inequality always holds except for the case , which was already handled in our previous paper [1, Example 2.6]. We obtain an -line L whose intersection with X has no -points; this line corresponds to a pencil of hypersurfaces of degree d such that each of the distinct -members is smooth.…”

“…We remark that the case n = 2 and d = 2 in Question 1.2 was investigated in our previous work [1,Example 2.6]. In that example, the smooth conics D 1 , D 2 , .…”

Existence of pencils with nonblocking hypersurfaces

“…We remark that linear systems of hypersurfaces over finite fields have been investigated from a few different perspectives in the literature (see for example [Bal07] and [Bal09]). There is also a version of "simultaneous" Bertini's theorem for a pencil of hypersurfaces over finite fields [AG22].…”

Tangent-Filling Plane Curves Over Finite Fields

“…S. Asgarli and D. Ghioca [2] (d) We say that C is tangent-filling if every point P ∈ P 2 (F q ) lies on a tangent line T Q C to the curve C at some smooth F q -point Q.…”