Morally speaking, my question is whether every real smooth projective curve can be deformed in as many real directions as complex directions. Let me make the question precise.
Let $H$ be the Hilbert scheme of curves of degree $d$ and genus $g$ in $\mathbb{P}^n$. Then real points $p\in H(\mathbb{R})$ correspond to curves defined over $\mathbb{R}$. Now assume that $p\in H(\mathbb{R})$ is a point that corresponds to a smooth irreducible curve and let $U\subset H(\mathbb{R})$ be an open neighbourhood (in the euclidean topology) of $p$.
Question 1. Is it true that $U$ is Zariski dense in the irreducible component of $H$ containing $p$?
The answer is "yes" if $p$ is a smooth point of $H$ but we know that even if $p$ corresponds to a smooth curve, it is not necessarily a smooth point of $H$. However, for our purposes it is already sufficient if $p$ is a smooth point of $H_{\textrm red}$ (reduced scheme associated to $H$). So a positive answer to my question would follow from a positive answer to this question which can be phrased over an arbitrary ground field:
Question 2. If $p\in H$ corresponds to a smooth and irreducible curve, is $p$ a smooth point of $H_{\textrm red}$?
