Linear system of hypersurfaces passing through a Galois orbit

Consider the vector space (parameter space) of all homogeneous forms of degree d in n+1 variables defined over some field K. Geometrically, the vanishing set of such a form corresponds to a hypersurface of degree d in the projective space P^{n}. The dimension of this parameter space is $m = \binom{n+d}{d}$. If P_1, ..., P_m are in "general position", then no hypersurface of degree d can pass through all these m points, because passing through each additional point imposes 1 new linearly independent condition. In this talk, we address the following variant: for a given K, d, and n, can we always find m points P_1, ..., P_m so that:

(a) P_1, P_2 ..., P_m form a Gal(L/K)-orbit of a single point P defined over a Galois extension L / K with [L:K] = m,  and

(b) No hypersurface of degree m defined over K passes through P_1, P_2, ..., P_m.

We show that the answer is "Yes" if the base field K has at least 3 elements. In other words, the concept of "general position" for points can be modelled by Galois orbits. As an application, we compute the maximum dimension of a linear system of hypersurfaces over a finite field F_q where each F_q-member of the system is irreducible over F_q. This is joint work with Dragos Ghioca and Zinovy Reichstein.