Density for the non-singular points in real varieties

Question

When I am checking "Singular point of an algebraic variety" from wikipedia, it says

"It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both open and dense in the variety (for the Zariski topology, as well as, $\textbf{in the case of varieties defined over the complexes, for the usual topology}$)"

My question is: When varieties are defined over the $\textbf{reals}$, is that true non-singular real points form a set that is dense in the real variety for the usual (real) topology?

Answer 1

It depends on what you mean by a variety defined over the reals.

If you mean $\operatorname{Spec} A$ where $A$ is an $\Bbb R$-algebra (or something covered by such affine opens), then the answer is yes by the standard scheme-theoretic methods - the set of singular points has at least codimension one in every irreducible component and its complement is therefore dense.

If you mean a set of the form $\{(x_1,\cdots,x_n)\in\Bbb R^n \mid f_1(x_1,\cdots,x_n) = \cdots = f_m(x_1,\cdots,x_n) = 0 \}$, the answer is no. Consider the Whitney umbrella, the surface in $\Bbb R^3$ defined by $x^2-y^2z=0$. The $z$-axis belongs to this variety, and is singular, and there are no points with $z<0$ in the umbrella but not on the $z$-axis. On the other hand, in the usual topology, no point with $z<0$ can be in the closure of a set where every point has $z\geq0$, so the negative $z$-axis is not in the closure of the smooth points of the variety in the usual topology.

Answer 2

The answer to your question is yes. The reason is simple. Because the usual topology is finer than the Zariski topology (It means that there are more open subsets in the usual topology than in the Zariski topology), a subset that is Zariski dense is automatically dense in the usual topology.

There is more to say about the nonsingular points of a real algebraic variety. For example, if the (Krull) dimension of the real algebraic variety is $d$, then the nonsingular points of this variety form a real analytic manifold of dimension $d$.

An easy to follow reference is Milnor's book "Singular Points of Complex Hypersurfaces". You may check Lemma 2.2, Theorem 2.3 and 2.4.