Let $(X,o)$ be an affine, isolated, normal, Gorenstein singularity. Let $f$ and $g$ be two morphisms from $\mbox{Spec}(\mathbb{C}[[t]])$ to $X$ (also known as formal arcs) such that the closed point maps to the singular point $o$ and the generic point maps to a smooth point of $X$. Denote by $C_f$ and $C_g$ the closure of the scheme-theoretic image of $f$ and $g$, respectively. Is the delta invariant of $C_f$ the same as that of $C_g$?
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1 Answer
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Without any stronger hypothesis connecting $f$ and $g$ you should not expect that to happen.
The simplest instance is when $(X,o)$ is alrealdy non singular, say dimension 2: $(\mathbb{C}^2,o)$. In this case the morphisms correspond to $\mathbb{C}$-algebra homomorphisms $\mathbb{C}\{x,y\}\rightarrow \mathbb{C}[[t]]$, and these can be interpreted as parametrizations of branches of plane curve singularities.
Take $f$ given by $x\mapsto t$, $y\mapsto t^k$ parametrizing a smooth branch ($\delta_f=0$) and $g$ given by $x\mapsto t^2$, $y\mapsto t^3$ a cusp ($\delta_g=1$).
So in general the answer is no,
