Contact loci in arc spaces

Jet Spaces and Arc Spaces

Before getting into details, I would like to recall the p-adic inspiration for this theory. Here is a list of things we looked at in Chapter III. (1) Polynomials f ∈ Z[X1, . . . , Xn] and their solutions over Z/pZ ' Zp/pZp, where p is a fixed prime and m ≥ 0. Such a solution can be written as x1 = a10 + a11p+ . . .+ a1mp m , . . . , xn = an0 + an1p+ . . .+ anmp m with ai ∈ {0, . . . , p− 1}. (2...

Arc Spaces , Motivic Integration

The concept ofmotivic integrationwas invented by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. He constructed a certain measure on the arc space of an algebraic variety, the motivic measure, with the subtle and crucial property that it takes values not in R, but in the Grothendieck ring of algebraic varieties. A whole theory on this subject wa...

Arc Spaces and Equivariant Cohomology

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex G-variety X by its associated arc space J∞X, with its induced G-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplic...

Stretchability of Jordan Arc Contact Systems

Geometry on arc spaces of algebraic varieties

This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.