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Families of jets through singularities of algebraic varieties are herestudied in relation to the families of arcs originally studied by Nash. Afterproving a general result relating them, we look at normal locally completeintersection varieties with rational singularities and focus on a class ofsingularities we call "higher Du Val singularities", a higher dimensional (andcodimensional) version of Du Val singularities that is closely related toArnold singularities. More generally, we introduce the notion of "highercompound Du Val singularities", whose definition parallels that of compound DuVal singularities. For such singularities, we prove that there exists aone-to-one correspondence between families of arcs and families of jets ofsufficiently high order through the singularities. In dimension two, the resultpartially recovers a theorem of Mourtada on the jet schemes of Du Valsingularities. As an application, we give a solution of the Nash problem forhigher Du Val singularities.

Families of jets of arc type and higher (co)dimensional Du Val singularities