We define graded, quasi-coherent O_S-algebras over a given base derived scheme S, and show that these are equivalent to derived G_m,S-schemes which are affine over S. We then use this G_m,S-action to define the projective spectrum Proj(B) of a graded algebra B as a quotient stack, show that Proj(B) is representable by a derived scheme over S, and describe the functor of points of Proj(B) in terms of line bundles. The theory of graded algebras and projective spectra is then used to define the blow-up of a closed immersion of derived schemes. Our construction will coincide with the existing one for the quasi-smooth case. The construction is done by generalizing the extended Rees algebra to the derived setting, using Weil restrictions. We close by also generalizing the deformation to the normal cone to the derived setting.

We define the derived blow-up Bl_ZX of a closed immersion Z → X of derived stacks, and give the deformation to the normal bundlefor any morphism X → Y of derived stacks which admits a cotangentcomplex. The derived blow-up is defined in terms of virtual Cartier divisors, and the deformation to the normal bundle is induced by the deformation space D_X/Y. The latter is defined via derived Weil restrictions. For a closed immersion Z → X, the projection D_Z/X → X × [A¹/G_m]is affine, which gives us the derived, extended Rees algebra R_Z/X^ext .The blow-up construction and the deformation to the normal bundle generalizes the quasi-smooth case from [KR19]. We show that Bl_ZX is the projective spectrum of R_Z/X, which is the construction used in [Hek21]. This comparison implies that Bl_ZX → X is schematic, and that the classical blow-up of Z_cl → X_cl is the schematic closure in (Bl_ZX)_cl of the open complement of (E_ZX)cl, where E_ZX is the derived exceptional divisor. The paper also includes an algebraicity and a smooth descent result for the deformation space. This is based on a detailed study of the derived Weil restriction along any affine morphism X → Y of finite Tor-amplitude, which can be of independent interest. 

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.

The 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 derived algebraic geometry, the second main example of a geometric context will be derived analytic geometry. The latter is a recent construction, and includes many different flavours of analytic geometry, such as complex analytic geometry, non-archimedean rigid analytic geometry, and analytc 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.

We construct a canonical stabilizer reduction X˜ for any derived 1-algebraic stack X over C as a sequence of derived Kirwan blow-ups, under mild natural conditions that include the existence of a good moduli space for the classical truncation X_cl. Our construction naturally generalizes Kirwan's classical partial desingularization algorithm to the context of derived algebraic geometry.

We prove that X˜ is a natural derived enhancement of the intrinsic stabilizer reduction constructed by Kiem, Li and the third author. Moreover, if X is (−1)-shifted symplectic, we show that the semi-perfect and almost perfect obstruction theory and their induced virtual fundamental cycle and virtual structure sheaf of X_cl˜, constructed by the same authors, are naturally induced by X˜ and its derived tangent complex. As a corollary, we give a fully derived perspective on generalized Donaldson-Thomas invariants of Calabi-Yau threefolds and define new generalized Vafa-Witten invariants for surfaces via Kirwan blow-ups.