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Computing the adjunction divisor

For any (closed) singular place we determine the multiplicity $ d_{[Q]}:=ord_Q ({\mathcal{A}})$, alternatively, by the formula (2) or the Dedekind formula (cf. Remarks 2.5 and 2.3). Note that to compute a local intersection number $ i_P(f,g)$ (as appearing in both formulas), we can proceed inductively, computing the primitive parametrization $ \varphi:K[[x,y]]\to
K[[t]]$ of $ g$ up to degree $ k$, until $ ord_t(\varphi(x){\:\text{mod}\:}
t^k,\varphi(y){\:\text{mod}\:} t^k)$ is less than $ k/ord(f)$.

Christoph Lossen