The space of arcs of an algebraic variety

The Masterpieces of John Forbes Nash Jr.

“…In his paper he formulated a question which became known in algebraic geometry as Nash's problem. A complete solution of the problem has not yet been reached although a lot of progress has been made in recent years (we refer the reader to the very recent survey [21]). Nash's problem (and his ideas) are nowadays formulated for varieties (in fact, schemes) on a general algebraically closed field of any characteristic.…”

“…3 An interesting case is that where W = V s is the set of singularities of V : X(V s ) consists of those arcs which "pass through" a singularity. In [78] Nash realized that this space has, roughly speaking, the structure of an "infinite dimensional complex variety" (for a precise formulation we refer to [78, p. 32] or to [21,Th. 2.6]; see also the earlier work of Greenberg [36]) which has finitely many irreducible components, cf.…”

“…Nonetheless the studies on Nash's problem are very far from being exhausted. Indeed the answer has been proved to be affirmative in a variety of interesting cases (see the survey articles [21,83]) and several mathematicians are looking for the "correct formulation" of the question (see, for instance, [55]), possibly leading to a complete understanding of the relations between resolutions of the singularities and the arc space.…”

“…In the modern literature it is customary to take an equivalent definition of X through formal power series; we refer to[57] for the latter and for several important subtleties related to variants of the Nash arc space 4. In fact, Nash claims the proposition with any algebraic subset W of V in place of V s but, although the proposition does hold for W = V s , it turns out to be false for a general algebraic subset W ; cf [21, . Ex.…”

The masterpieces of John Forbes Nash Jr

“…In his paper he formulated a question which became known in algebraic geometry as Nash's problem. A complete solution of the problem has not yet been reached although a lot of progress has been made in recent years (we refer the reader to the very recent survey [21]).…”