For any modification of a smooth variety by blow-ups at a point, there is a rational function whose prescribed exceptional components are exactly the dicritical ones with prescribed degrees.
Casas-Alvero,Approximate implicit equations of parameterized germs of plane curve, JP J
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Dicritical divisors and hypercurvettes
For any modification of a smooth variety by blow-ups at a point, there is a rational function whose prescribed exceptional components are exactly the dicritical ones with prescribed degrees.