# Geometry of curves with $A_n$ singularity

I am a beginner in the study of curves with $A_n$ singularities. I had the following questions: Fix $n \ge 1$ and $N \ge 3$

1) Fix $2$ integers $d, g$. Is it true that the set of curves of degree $d$ and arithmetic genus $g$ in $\mathbb{P}^N$ with an $A_n$ singularity can be seen as a subscheme of a Hilbert scheme?

2) If the answer to ($1$) is true, can the subscheme be an open subset of an irreducible component?

3) Is it very difficult to compute the dimension of such subschemes? Examples/ideas/references for such computations will be very interesting to see.

4) Can we say something similar for $D_n$ singularities?

Any suggestions for references on this topic is most welcome.

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Why the downvote? –  Naga Venkata Mar 18 at 17:13