10 The cross-sectional area of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ through the diagonal $B D_{1}$ is $S, S_{\max }$ and $S_{\min }$ are the maximum and minimum values of $S$, respectively. Then the value of $\frac{S_{\max }}{S_{\min }}$ is ( ).
(A) $\frac{\sqrt{3}}{2}$
(B) $\frac{\sqrt{6}}{2}$
(C) $\frac{2 \sqrt{3}}{3}$
(D) $\frac{2 \sqrt{6}}{3}$
$10 \mathrm{C}$ Prompt: The section is divided into 2 identical triangles by $B D_{1}$, meaning the ratio of heights with $B D_{1}$ as the base is the area ratio. The maximum height is $\frac{\sqrt{6}}{3}$, and the minimum height is $\frac{\sqrt{2}}{2}$, so
$$
\frac{S_{\max }}{S_{\min }}=\frac{2 \sqrt{3}}{3} .
$$