Blow-ups and normal bundles in connective and nonconnective derived geometries
Oren Ben-Bassat, Jeroen Hekking
TL;DR
The paper generalizes derived blow-ups and deformation to the normal bundle from derived algebraic geometry to arbitrary geometric contexts, organized around a fixed derived algebraic context $\mathscr{C}$. By developing $\mathscr{C}$-stacks and $\mathscr{C}_{\ge0}$-stacks, it constructs the deformation space $\mathrm{D}_{X/Y}$ and the extended Rees algebra $\mathcal{R}_{X/Y}^{\operatorname{ext}}$, allowing the blow-up to be defined as $\mathrm{Bl}_X Y = \operatorname{Proj}(\mathcal{R}_{X/Y})$ and to analyze connectivity via closed immersions. The deformation-to-the-normal-bundle framework identifies the normal data and connects to virtually Cartier divisors, showing that the blow-up classifies strict virtual Cartier divisors and, under finiteness, is nonconnectively affine over the base, with the construction compatible with base-change. The theory applies to both derived algebraic geometry and derived analytic geometry, including nonconnective contexts, thereby broadening the toolbox for intersection theory, moduli problems, and analytic geometry in derived settings.
Abstract
This work presents a generalization of derived blow-ups and of the derived deformation to the normal bundle from derived algebraic geometry to any geometric context. The latter is our proposed globalization of a derived algebraic context, itself a generalization of the theory of simplicial commutative rings. One key difference between a geometric context and ordinary derived algebraic geometry is that the coordinate ring of an affine object in the former is not necessarily connective. When constructing generalized blow-ups, this not only turns out to be remarkably convenient, but also leads to a wider existence result. Indeed, we show that the derived Rees algebra and the derived blow-up exist for any affine morphism of stacks in a given geometric context. However, in general the derived Rees algebra will no longer be connective, hence in general the derived blow-up will not live in the connective part of the theory. Unsurprisingly, this can be solved by restricting the input to closed immersions. The proof of the latter statement uses a derived deformation to the normal bundle in any given geometric context, which is also of independent interest. Besides the geometric context which extends algebraic geometry, the second main example of a geometric context will be an extension of analytic geometry. The latter is a recent construction, and includes many different flavors of analytic geometry, such as complex analytic geometry, non-archimedean rigid analytic geometry and analytic geometry over the integers. The present work thus provides derived blow-ups and a derived deformation to the normal bundle in all of these, which is expected to have many applications.