An affine spread is a set of subspaces of \(\textrm{AG}(n, q)\) of the same dimension that partitions the points of \(\textrm{AG}(n, q)\). Equivalently, an affine spread is a set of projective subspaces of \(\textrm{PG}(n, q)\) of the same dimension which partitions the points of \(\textrm{PG}(n, q) \setminus H_{\infty }\); here \(H_{\infty }\) denotes the hyperplane at infinity of the projective closure of \(\textrm{AG}(n, q)\). Let \(\mathcal {Q}\) be a non-degenerate quadric of \(H_\infty \) and let \(\Pi \) be a generator of \(\mathcal {Q}\), where \(\Pi \) is a t-dimensional projective subspace. An affine spread \(\mathcal {P}\) consisting of \((t+1)\)-dimensional projective subspaces of \(\textrm{PG}(n, q)\) is called hyperbolic, parabolic or elliptic (according as \(\mathcal {Q}\) is hyperbolic, parabolic or elliptic) if the following hold:

• Each member of \(\mathcal {P}\) meets \(H_\infty \) in a distinct generator of \(\mathcal {Q}\) disjoint from \(\Pi \);

• Elements of \(\mathcal {P}\) have at most one point in common;

• If \(S, T \in \mathcal {P}\), \(|S \cap T| = 1\), then \(\langle S, T \rangle \cap \mathcal {Q}\) is a hyperbolic quadric of \(\mathcal {Q}\).

In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of \(\textrm{PG}(n, q)\) is equivalent to a spread of \(\mathcal {Q}^+(n+1, q)\), \(\mathcal {Q}(n+1, q)\) or \(\mathcal {Q}^-(n+1, q)\), respectively.

仿射扩散是一组具有相同维度的 \(\textrm{AG}(n, q)\) 子空间，用于划分 \(\textrm{AG}(n, q)\) 的点。同样，仿射散布是一组相同维度的 \(\textrm{PG}(n, q)\) 射影子空间，它划分 \(\textrm{PG}(n, q) \setminus H_ 的点{\infty}\);这里 \(H_{\infty }\) 表示 \(\textrm{AG}(n, q)\) 射影闭包的无穷远超平面。令 \(\mathcal {Q}\) 为 \(H_\infty \) 的非简并二次曲线，并令 \(\Pi \) 为 \(\mathcal {Q}\) 的生成元，其中 \(\ Pi \) 是一个 t 维射影子空间。由 \(\textrm{PG}(n, q)\) 的 \((t+1)\) 维射影子空间组成的仿射展开 \(\mathcal {P}\) 称为双曲、抛物线或椭圆 (根据 \(\mathcal {Q}\) 是双曲形、抛物线形或椭圆形）如果以下成立：

• 
  \(\mathcal {P}\) 的每个成员在与 \(\Pi \) 不相交的 \(\mathcal {Q}\) 的不同生成器中满足 \(H_\infty \)；

• 
  \(\mathcal {P}\) 的元素至多有一个共同点；

• 
  如果 \(S, T \in \mathcal {P}\), \(|S \cap T| = 1\)，则 \(\langle S, T \rangle \cap \mathcal {Q}\) 是\(\mathcal {Q}\) 的双曲二次曲线。

在本注释中，表明 \(\textrm{PG}(n, q)\) 的双曲、抛物线或椭圆仿射展开等价于 \(\mathcal {Q}^+(n+1, q)\)、\(\mathcal {Q}(n+1, q)\) 或 \(\mathcal {Q}^-(n+1, q)\) 分别。