On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| ≥ 4 and n, m ≥ 2, or |k| = 3 and n ≥ 3, m ≥ 2. Then, for any embedding ψ of AG(n, k) into PG(m, K), there exists an isomorphism θ from k into K and an (n+1) × (m+1) matrix B with entries in K such that ψ can be expressed as ψ (x1,x2,...,xn) = [(1,x1θ ,x2θ ,...,xnθ)B], where the right-hand side is the equivalence class of (1,x1θ ,x2θ,...,xnθ)B. Moreover, in this expression, θ is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l ≥ 1, and suppose that there exists an embedding ψ of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dimkθK, then we have r ≥ 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding ψ of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of kθ with dim kθK=r ≥ 3, and if m ≥ 2l/(r-2) with m even or if m ≥ 2l/(r-2) +1 with m odd.