Derived jet and arc spaces

TL;DR

This paper explores derived jet and arc spaces in algebraic geometry, connecting classical and derived perspectives, and introduces new invariants and applications for singular spaces.

Contribution

It develops the theory of derived jet and arc spaces, generalizes formulas for cotangent complexes, and extends key results like Reguera's lemma to broader contexts.

Findings

Derived constructions match classical ones for smooth schemes.

New singularity invariants via higher homotopy groups.

Extended arc space results to non-perfect base fields.

Abstract

We study jet schemes and arc spaces in the context of derived algebraic geometry. Explicitly, we consider the jet and arc functors in the category of schemes and study their animations to the category of derived schemes -- what we call the derived jet and arc spaces. We show that the derived constructions agree with the classical versions when the base scheme is smooth, or more generally for local complete intersection log canonical singularities, giving a derived interpretation to a theorem of Musta\c{t}\u{a}. For more singular spaces we get new singularity invariants in the form of higher homotopy groups. We also study cotangent complexes for derived jet and arc spaces, generalizing previous formulas for sheaves of differentials of classical jet and arc spaces. Several applications are obtained. Specifically, we revisit recent results on the local structure of arc spaces from the lens of cotangent complexes, giving more unified proofs and removing unnecessary hypotheses. In particular, we extend a version of Reguera's curve selection lemma for arc spaces to the case of non-perfect base fields.