The purpose of this note is to correct a mistake in the article “A curve selection lemma in spaces of arcs and the image of the Nash map” Compositio Math. 142 (2006), 119–130. It is due to an overlooked hypothesis in the definition of generically stable subset of the space of arcs X∞ of a variety X defined over a perfect field k.

The Nash problem asks about the existence of a correspondence between families of arcs through singularities of complex varieties and certain types of divisorial valuations. It has been positively settled in dimension 2 by Fernández de Bobadilla and Pe Pereira, and it was shown to have a negative answer in all dimensions ${\geq }4$ by Ishii and Kollár. In this note we discuss examples which show that the problem has a negative answer in dimension 3 as well. These examples also bring to light the different nature of the problem depending on whether it is formulated in the algebraic setting or in the analytic setting.

We prove that, if X is a variety over an uncountable algebraically closed field k of characteristic zero, then any irreducible exceptional divisor E on a resolution of singularities of X which is not uniruled, belongs to the image of the Nash map, i.e. corresponds to an irreducible component of the space of arcs on X centred in Sing X. This reduces the Nash problem of arcs to understanding which uniruled essential divisors are in the image of the Nash map, more generally, how to determine the uniruled essential divisors from the space of arcs.

We prove a finiteness result in the space of arcs $X_\infty$ of a singular variety X, which is an extension of the stability result of Denef and Loeser (Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232). From this follows a curve selection lemma for generically stable subsets of $X_\infty$. As a consequence, we extend to all dimensions the problem of wedges, proposed by Lejeune-Jalabert (Arcs analytiques et résolution minimale des singularités des surfaces quasi-homogénes, Lecture Notes in Mathematics, vol. 777 (Springer, Berlin, 1980), 303–336), and we obtain that an affirmative answer to this problem is equivalent to the surjectivity of the Nash map. This implies, for instance, that the Nash map is bijective for sandwiched surface singularities.

Buddington (1959) pointed out that the construction of large crustal magma chambers involves complex internal processes as well as multiple country rock material transfer processes (MTPs), which reflect large horizontal, vertical and temporal gradients in physical conditions. Thus, we have attempted to determine the relative importance of different magmatic and country rock MTPs at various crustal depths, and whether country rock MTPs largely transport material vertically or horizontally, rather than seeking a single model of magma ascent and emplacement.

Partially preserved roofs of nine plutons and in some cases roof–wall transitions with roof emplacement depths of 1·5–11 km were mapped. During emplacement, these roofs were not deformed in a ductile manner, detached or extended by faults, or significantly uplifted. Instead, sharp, irregular, discordant contacts are the rule with stoped blocks often preserved immediately below the roof, even at depths of 10 km. The upper portions of these magma chambers are varied, sometimes preserving the crests of more evolved magmas or local zones of volatile-rich phases and complex zones of dyking and magma mingling. Magmatic structures near roofs display a wide variety of patterns and generally formed after emplacement. Transitions from gently dipping roofs to steep walls are abrupt. At shallow crustal levels, steep wall contacts have sharp, discordant, stepped patterns with locally preserved stoped blocks indicating that the chamber grew sideways in part by stoping. Around deeper plutons, an abrupt transition (sometimes within hundreds of metres) occurs in the country rock from discordant, brittle roofs to moderately concordant, walls deformed in a ductile manner defining narrow structural aureoles. Brittle or ductile faults are not present at roof–wall joins.

Near steep wall contacts at shallow to mid-crustal depths (5–15 km), vertical and horizontal deflections of pre-emplacement markers (e.g. bedding, faults, dykes), and ductile strains in narrow aureoles (0·1–0·3 body radii) give a complete range of bulk strain values that account for 0–100% of the needed space, but average around 30%, or less, particularly for larger batholiths. A lack of far-field deflection of these same markers rules out significant horizontal displacement outside the aureoles and requires that any near-field lateral shortening is accommodated by vertical flow. Lateral variations from ductile (inner aureole) to brittle (outer aureole) MTPs are typically observed. Compositional zoning is widespread within these magma bodies and is thought to represent separately evolved pulses that travelled up the same magma plumbing system. Magmatic foliations and lineations commonly cross-cut contacts between pulses and reflect the strain caused either by the late flow of melt or regional deformation.

Country rocks near the few examined mid- to deep crustal walls (10–30 km) are extensively deformed, with both discordant and concordant contacts present; however, the distinction between regional and emplacement-related deformation is less clear than for shallower plutons. Internal sheeting is more common, although elliptical masses are present. Lateral compositional variations are as large as vertical variations at shallower depths and occur over shorter distances. Magmatic foliations and lineations often reflect regional deformation rather than emplacement processes.

The lack of evidence for horizontal displacement outside the narrow, shallow to mid-crustal aureoles and the lack of lateral or upwards displacement of pluton roofs indicate that during emplacement most country rock is transported downwards in the region now occupied by the magma body and its aureole. The internal sheeting and zoning indicate that during the downwards flow of country rock, multiple pulses of magma travelled up the same magma system. If these relationships are widespread in arcs, magma emplacement is the driving mechanism for a huge crustal-scale exchange process.