Let $H = Hilb_{d,g,r}$ be the Hilbert scheme of genus $g$ curves of degree $d$ in proyective space $\mathbb{P}^r$.

Is it true that the set of points of $H$ consisting of smooth curves is open in $H$? If so, how do I see this?

Let $H = Hilb_{d,g,r}$ be the Hilbert scheme of genus $g$ curves of degree $d$ in proyective space $\mathbb{P}^r$, over an algebraically closed field $k$.

Is it true that the set of points of $H$ consisting of smooth curves is open in $H$? If so, how do I see this?

Edit: added the hypothesis that the base field is algebraically closed.