According to the book "moduli of curves" by Harris. He defined $U_{d,g}$ as the **variety** in $\mathbb{P}^N$ (coefficient space) of (irreducible reduced) nodal curves with genus $g$ and degree $d$. We first ignore the other difficult issues about the irreducibility. On page 30 theorem 1.49, he claimed:

$U_{d,g}$ is smooth of dimension $3d+g-1$.

First, let's simplify the argument of one node. He proved like this: Take $\Sigma=\{(C,p)|p \text{ is singular on }C\}\subseteq \mathbb{P}^N\times\mathbb{P}^2$. The proof is a bit mixed with differential geometry: write down the polynomials $$F(X,Y)=0\\F_x(X,Y)=0\\F_y(X,Y)=0$$

and apply constant rank theorem. These are all fine, and we can show that $\Sigma$ is smooth of dimension $N-1$ at those curves with only one nodes.

Now, he takes projection $\pi:\Sigma\to\mathbb{P}^n$. In order to the desired image we must take away those diagonals in order to forbid repeating nodes. Then the map is an immersion at each $(C,p)$ and the image is exactly $U_{d,g}$. Here comes the problem. We can only be sure that the image is a constructible set. Even though $\pi$ is an injective immersion, we can not conclude that the image is smooth (manifold) because of this.

Do I miss something? Actually, I tried to prove that this map is an embedding. But the map can not be proved to be a homeomorphism.