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.
Arnol′d, Singularities of fractions and the behavior of polynomials at infinity, Tr
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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.