Deformation to the normal bundle and blow-ups via derived Weil restrictions

TL;DR

This paper introduces a derived algebraic geometry framework for deformation to the normal bundle and blow-ups, utilizing derived Weil restrictions to generalize classical constructions to a derived setting.

Contribution

It develops a derived analogue of deformation to the normal cone and extends derived blow-up theory to arbitrary centers, broadening previous quasi-smooth results.

Findings

Constructed a derived deformation to the normal cone via derived Weil restriction.

Proved algebraicity of derived Weil restrictions along finite morphisms.

Generalized derived blow-ups to arbitrary closed centers.

Abstract

We develop an analogue of the deformation to the normal cone in the context of derived algebraic geometry. This provides any given morphism of derived stacks with a degeneration to the zero section of its normal bundle (i.e., its 1-shifted relative tangent bundle). The construction is realized via the derived Weil restriction along the zero section of the affine line. We prove a general algebraicity theorem for derived Weil restrictions along finite but possibly non-flat morphisms. As an application of the theory, we study derived blow-ups along arbitrary closed centres, generalizing previous works of the authors in the quasi-smooth case.