In this paper, we consider the parameter space of the rationalplane curves with uni-branched singularity. We show that such a parameterspace is decomposable into irreducible components which are rationalvarieties. Rational parametrizations of the irreducible componentsare given in a constructive way, by a repeated use of Abhyankar-MohEpimorphism Theorem. We compute an enumerative invariant of thisparameter space, and include explicit computational examples to recoversome classically-known invariants.

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