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

de Fernex, Tommaso; Wang, Shih-Hsin

Comptes Rendus. Mathématique, Complex algebraic geometry, in memory of Jean-Pierre Demailly, Tome 362 (2024), pp. 119-139

Families of jets through singularities of algebraic varieties are here studied in relation to the families of arcs originally studied by Nash. After proving a general result relating them, we look at normal locally complete intersection varieties with rational singularities and focus on a class of singularities we call higher Du Val singularities, a higher dimensional (and codimensional) version of Du Val singularities that is closely related to Arnold singularities. More generally, we introduce the notion of higher compound Du Val singularities, whose definition parallels that of compound Du Val singularities. For such singularities, we prove that there exists a one-to-one correspondence between families of arcs and families of jets of sufficiently high order through the singularities. In dimension two, the result partially recovers a theorem of Mourtada on the jet schemes of Du Val singularities. As an application, we give a solution of the Nash problem for higher Du Val singularities.

Les familles de jets à travers les singularités des variétés algébriques sont étudiées ici en relation avec les familles d’arcs initialement étudiées par Nash. Après avoir démontré un résultat général les concernant, nous examinons les variétés d’intersection localement complètes normales avec des singularités rationnelles et nous concentrons sur une classe de singularités que nous appelons « singularités de Du Val supérieures » , une version de dimension (et codimension) supérieure des singularités de Du Val étroitement liée aux singularités d’Arnold. Plus généralement, nous introduisons la notion de « singularités de Du Val composées supérieures » , dont la définition est parallèle à celle des singularités de Du Val composées. Pour de telles singularités, nous démontrons qu’il existe une correspondance bijective entre les familles d’arcs et les familles de jets d’ordre suffisamment élevé à travers les singularités. En dimension deux, le résultat récupère partiellement un théorème de Mourtada sur les schémas de jets des singularités de Du Val. En tant qu’application, nous proposons une solution au problème de Nash pour les singularités de Du Val supérieures.

Classification : 14E18, 14B05
Keywords: Jet scheme, arc space, Nash problem, rational singularity
Mots-clés : Schémas de jet, espace d’arc, problème de Nash, singularité rationnelle

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