[Paper Review] Exceptional divisors which are not uniruled belong to the image of the Nash map

Monique Lejeune-Jalabert, Ana J. Reguera|ArXiv.org|Nov 14, 2008

Algebraic Geometry and Number Theory9 references15 citations

TL;DR

This paper proves that over an uncountable algebraically closed field of characteristic zero, any irreducible exceptional divisor on a resolution of a singular variety that is not uniruled must belong to the image of the Nash map—meaning it corresponds to an irreducible component of the space of arcs centered at the singular locus. The result reduces the Nash problem to understanding which uniruled essential divisors lie in the image of the Nash map, using wedge lifting and Lüroth's theorem.

ABSTRACT

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 centered 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.

Motivation & Objective

• To resolve part of the Nash problem by determining which essential divisors arise from irreducible components of the space of arcs centered at the singular locus.
• To characterize the image of the Nash map using wedge lifting properties and stable points in the arc space.
• To show that non-uniruled essential divisors are always in the image of the Nash map, thereby reducing the full surjectivity question to uniruled divisors.
• To provide a criterion for membership in the image of the Nash map based on lifting of wedges from closed points in a locally closed subset of the arc space.
• To connect the surjectivity of the Nash map for surface singularities to the wedge-lifting property of resolutions of quasirational singularities.

Proposed method

• Introduces a lifting criterion for wedges centered at closed points of a locally closed subset of the arc space associated to an essential divisor.
• Uses Lüroth’s theorem to deduce that non-uniruled divisors correspond to components of the arc space via elimination of indeterminacy in rational maps from wedges.
• Applies the curve selection lemma for stable points in the arc space, as developed by Reguera, to analyze the geometry of arcs and their centers.
• Constructs formal completions of schemes along curves to lift $Ψ$-maps from the arc space to formal neighborhoods of singular points.
• Utilizes the theorem on formal functions to relate inverse limits of sheaf cohomology to formal completions, ensuring the existence of wedge lifts.
• Relies on the existence of resolutions of singularities where exceptional divisors are rational curves, particularly in the case of surface singularities.

Experimental results

Research questions

• RQ1 Does every essential divisor over a variety in characteristic zero that is not uniruled belong to the image of the Nash map?
• RQ2 Can the surjectivity of the Nash map be reduced to analyzing only uniruled essential divisors?
• RQ3 What conditions on a resolution ensure that a given essential divisor arises from an irreducible component of the arc space centered at the singular locus?
• RQ4 How do wedge lifting properties relate to the image of the Nash map in the context of essential divisors?
• RQ5 Under what conditions does a resolution of a quasirational surface singularity allow lifting of wedges, and how does this affect the surjectivity of the Nash map?

Key findings

• Any irreducible exceptional divisor on a resolution of a variety over an uncountable algebraically closed field of characteristic zero that is not uniruled belongs to the image of the Nash map.
• The surjectivity of the Nash map for normal surface singularities over $\mathbb{C}$ would follow if every quasirational surface singularity admits a resolution with the wedge-lifting property for each essential divisor.
• The image of the Nash map is completely determined by the lifting of wedges from closed points in a locally closed subset of the arc space associated to each essential divisor.
• The construction of wedge lifts relies on formal completion and the theorem on formal functions, ensuring that morphisms from formal neighborhoods lift from the arc space.
• The result reduces the full Nash problem to the case of uniruled essential divisors, as non-uniruled ones are already in the image.
• The paper provides a criterion for membership in the image of the Nash map based on the existence of wedge lifts from closed points, generalizing earlier results on stable points.