We give a theory on the space of arcs $X_\infty$
of a singular variety $X$, based on the finiteness property of the
stable points of $X_\infty$. We also show how computations can be
made in this theory. As a consequence, we obtain a process of
determining invariants of $X$ from its space of arcs $X_\infty$,
and from its spaces of wedges (arcs in the space of arcs). More
precisely, we give two concepts of regularity for the stable
points of $X_\infty$, and we show that we have regularity in
codimension $1$, and for all stable points of the space of arcs of
a curve. As another application, we show concrete examples, obtained
from the spaces of arcs, of Noetherian $1$-dimensional local rings
which are analytically
ramified.

Towards the singular locus of the space of arcs

We study arc spaces and jet schemes of generic determinantal varieties. Using the natural group action, we decompose the arc spaces into orbits, and analyze their structure. This allows us to compute the number of irreducible components of jet schemes, log canonical thresholds, and topological zeta functions.

/pdf/arcs-on-determinantal-varieties-3ss3l3ck17.pdf

Arcs on determinantal varieties

We prove that Nash mapping is bijective for any algebraic surface defined over an algebraically closed field of characteristic 0.

Nash problem for surfaces

This paper is an introduction to the jet schemes and the arc space of an algebraic variety We also introduce the Nash problem on arc families Resume Ce papier constitue une introduction aux espaces de jets etl'espace d'arcs d'une variete algebrique Nous introduisons ´egalement le probleme de Nash pour les familles d'arcs

Jet schemes, arc spaces and the nash problem

Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers- Ramanujan Identities. The linking object is the Hilbert-Poincar´ e series of the arc space over a point of the base variety. In the case of the double point this is precisely the generating series for the integer partitions without equal or consecutive parts. R´ esum´

http://combinatorics.cis.strath.ac.uk/fpsac2011/proceedings/dmAO0120.pdf

Arc Spaces and Rogers-Ramanujan Identities

(Une classe ·de singularites non-rationnelles de surfaces ayant une application de Nash bijective) Soit (S, 0) un germe de surface analytique complexe normale. Nous considerons le diviseur exceptionnel reduit E et ses composantes irreductibles E i ∈I, sur sa resolution minimale. L'application de Nash associe a chaque composante irreductible C k de l'espace des arcs passant par 0 sur S, l'unique composante de E rencontree par la transformee stricte de l'arc generique dans C κ . Nash a prouve son injectivite et a demande si elle etait bijective. Nous prouvons que c'est le cas si E E i < 0 pour tout i ∈ I comme cas particulier de notre theoreme principal.

/pdf/a-class-of-non-rational-surface-singularities-with-bijective-4zk62uvqh4.pdf

A class of non-rational surface singularities with bijective Nash map

Let (X,0) be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over (X,0) is a divisorial valuation of the field of meromorphic functions on (X,0), whose center on any resolution of the germ is an irreducible component of the exceptional locus. The Nash map associates to each irreducible component of the space of arcs through 0 on X the unique essential divisor intersected by the strict transform of the generic arc in the component. Nash proved its injectivity and asked if it was bijective. We prove that this is the case if there exists a divisorial resolution \pi of (X,0) such that its reduced exceptional divisor carries sufficiently many \pi-ample divisors (in a sense we define). Then we apply this criterion to construct an infinite number of families of 3-dimensional examples, which are not analytically isomorphic to germs of toric 3-folds (the only class of normal 3-fold germs with bijective Nash map known before).

Families of higher dimensional germs with bijective Nash map

Let (X,0) be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over (X,0) is a divisorial valuation of the field of meromorphic functions on (X,0), whose center on any resolution of the germ is an irreducible component of the exceptional locus. The Nash map associates to each irreducible component of the space of arcs through 0 on X the unique essential divisor intersected by the strict transform of the generic arc in the component. Nash proved its injectivity and asked if it was bijective. We prove that this is the case if there exists a divisorial resolution π of (X,0) such that its reduced exceptional divisor carries sufficiently many π-ample divisors (in a sense we define). Then we apply this criterion to construct an infinite number of families of 3-dimensional examples, which are not analytically isomorphic to germs of toric 3-folds (the only class of normal 3-fold germs with bijective Nash map known before).

/pdf/families-of-higher-dimensional-germs-with-bijective-nash-map-h5lix8psnk.pdf

Camille Plenat

Papers

A class of non-rational surface singularities with bijective Nash map

Families of higher dimensional germs with bijective Nash map

Families of higher dimensional germs with bijective Nash map

A class of non-rational surface singularities for which the Nash map is bijective

Resolution du probleme des arcs de Nash pour les points doubles rationnels