After reading this question on MO on the "degree" of an affine variety, I did not see why they said it was hard to give a lower bound on the number of intersection points of an affine variety $X \subseteq \mathbb{A}^n$ with an $(n - \mathrm{dim}(X))$-dimensional linear subspace in general position. In particular, I would expect these to be the same in the affine and projective case. Could anyone explain this?
Details
Let us define the "degree" $\deg(X)$ of an affine variety $X$ (with implicit embedding $X \to \mathbb{A}^n$) as the number of points in the intersection of $X$ with an $(n - \dim(X))$-dimensional linear subspace of $\mathbb{A}^n$ in general position to $X$. This is analogous to the projective case.
Then I would expect that $\deg_{\mathbb{A}^n}(X) = \deg_{\mathbb{P}^n}(\overline{X})$, where $\overline{X}$ is the projective closure of $X \subseteq \mathbb{A}^n \subseteq \mathbb{P}^n$.
Proof Sketch Clearly the projective closure $\overline{X}$ of $X$ does not have any irreducible components lying in the hyperplane at infinity $H := \mathbb{P}^n \setminus \mathbb{A}^n$. Hence, the projective variety $Y := \overline{X} \cap H \subseteq H \subseteq \mathbb{P}^n$ has dimension $d - 1$, where $d = \dim(\overline{X}) = \dim(X)$.
Now, a $(n - d)$-dimensional linear subspace $L$ of $\mathbb{P}^n$ in general position clearly does not intersect $Y$, as $\dim(L) + \dim(Y) = n - 1 < n = \dim(\mathbb{P}^n)$ (the statement $L$ intersects $Y$ is a nontrivial, algebraically closed condition on $L$ in $\mathrm{Gr}(d - n, \mathbb{P}^n))$. However, this should give \begin{align*} \deg_{\mathbb{P}^n}(\overline{X}) &= \#(L \cap \overline{X}) = \#(L \cap Y) + \#(L \cap X) \\ &= \#(L \cap X) = \#((L \cap \mathbb{A}^n) \cap X) = \deg_{\mathbb{A}^n}(X) \end{align*} since $L \cap \mathbb{A}^n$ is clearly a $(n - d)$-dimensional linear subspace of $\mathbb{A}^n$ in general position. $\square$
I know that I am somewhat imprecise with my use of ``general position'', but intuitively, this seems to make sense. However, the MO question seems to indicate that the above does not hold in general, and we just have \begin{equation*} \deg_{\mathbb{A}^n}(X) \leq \deg_{\mathbb{P}^n}(\overline{X}) \end{equation*} Can anyone tell me where/whether the statement or my proof sketch are wrong?
Your proof sketch looks fine to me - from a close reading of the comments at the MO post, it looks like the commenter is concerned about answering the question "what are the bounds for $\#(L\cap X)$ depending on $X,L$", not "what happens for general $L$". I'd still caution you that there's a reason people don't use "degree in $\Bbb A^n$" - it's much less nice than the projective version.