Let $f(X,Y) \in \mathbb{Q}[X,Y]$ be a quadratic polynomial. The Hasse-Minkowski theorem says that $f(X,Y) = 0$ has a solution $(x,y) \in \mathbb{Q}^2$ iff it has a solution in $\mathbb{R}^2$ and $\mathbb{Q}^2 _p$ for every prime $p$, and this famously fails for higher-order forms like cubics, etc., with the failure measured by the Tate-Shafarevich group.
Is there a nice, simple geometric way to think about this?
Edit: More specifically, what does it "look like" when you piece together local solutions into a global solution? Is there any good way to visualize this?