Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big ?
Is there a formula relating the dimension of the Zariski tangent space and the order of the singularity?
What are the basic references (books) for learning about classification of complex curve singularities (not necessarily plane)?
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Another way to make a curve with many branches through a point is to start with $\mathbb A^1$ over $\mathbb C$ (say) and then glue together the points $t = 0, 1, \ldots, n$ (for some the value of $n$).
That is, we let $A$ be the $\mathbb C$-subalgebra of $\mathbb C[t]$ consisting of polynomials $f$ such that $f(0) = f(1) = \ldots = f(n).$ This a finitely generated $\mathbb C$-algebra corresponding to an (irreducible) curve with a singularity through which there are $n+1$ branches. A computation shows that the tangent space at the origin has dimension $n+1$.
Consider the curve singularity at the origin of the image $C$ of the map $z \mapsto (z^e, z^{e + 1} \ldots, z^{e + n})$ where $e$ is a large integer (say larger than $n + 1$). Any polynomial equation for $C$ must have vanishing constant and linear terms. Hence the Zariski tangent space of $C$ at $0$ has dimension $n + 1$.