Let $V$ be an irreducible complex plane algebraic curve, $V=V(f)$, and let $\mathcal{O}_p$ be the local ring of holomorphic functions defined in some neighborhood of $p$.

If $p=(0,0)$ is a smooth point then $f$ has linear terms hence $f$ must be irreducible in $\mathcal{O}_p$. Thus the local analytic curve at $p$ is irreducible.

$\textbf{Question 1:}$ Is this a sufficient condition for a point to be smooth? The normalization theorem tell us that if there is one local analytic curve component then a neighborhood of $p$ in $V$ is homeomorphic to a disk, but this does not guarantee that $p$ will be smooth. Indeed $p$ could be a cusp.

$\textbf{Question 2:}$ More generally, if $p$ is a $k$-tuple point, what can we say about the relationship between $k$ and the number of local analytic curve components? Does it matter if $p$ is ordinary, i.e., the $k$ tangent lines at $p$ are distinct?


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