This talk is about joint work with I. Swanson.

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A longstanding question in algebraic geometry is the classification of  reduced and 
irreducible local complete one--dimensional domains $R$ over an algebraically closed 
field $k$. It is known that such a ring is completely determined once it is known up 
to a "sufficiently high" power of its maximal ideal, where this sufficiently 
high power depends on the singularity degree $\delta$ of the ring.

In this talk we show that two curve singularities $(R, \mathfrak m)$ and $(R', \mathfrak m')$ 
are already isomorphic if there exists an isomorphishm 
$\varphi: R/ \mathfrak m^{j+1} \longrightarrow R'/ {\mathfrak m'}^{j+1}$ of 
$k$--algebras for some $j \geq 2 \delta +1$, and that the isomorphism may be chosen to agree with $\varphi 
\pmod{\mathfrak m^{j-2 \delta+1}}$. This strengthens a result of Hironaka, who obtained the
bound $3 \delta + 1$.