curves in varieties

Question

Let $V$ be an affine algebraic variety defined over $\mathbb R$. We assume that $V\subset \mathbb A_n$(affine $n$ space). Suppose that for any algebraic curve $C$ in $V$ defined over $\mathbb R$, the real points $C(\mathbb R)$ is contained in a finite union of proper affine $\mathbb R$-subspaces of $\mathbb A_n$. Is it true that $V(\mathbb R)$ is contained in a finite union of affine $\mathbb R $-subspaces of $\mathbb A_n$.

Answer 1

Yes. We can assume without loss of generality that the real points of $V$ are Zariski dense. Choose some linear projection $\rho: V \to \mathbb R^{\dim V}$ whose image includes some open ball, e.g. a generic linear projection. Then if $L$ is a linear subspace of codimension $1$ in $\mathbb R^n$, $\rho(V \cap L )$ is a variety of dimension at most $\dim V -1$. So we have a $n$-dimensional family of subvarieties of codimension $1$ of this open ball.

Choose a finite set of points of the open ball which is on none of these subvarieties.

Choose a generic curve of high degree passing through the finite set of points. Such a curve will have infinitely many real points, and be irreducible. Or just choose a parametrized curve passing through those points. Since it does not lie in $\rho(V \cap L)$, for any $L$, and is irreducible it intersects $\rho(V \cap L)$ at only finitely many points, so it is contained in no finite union.

Then the inverse image of this curve in $V$ will be contained in the union of no finite set of linear subspaces.