Blow-ups and normal bundles in derived algebraic geometry and beyond

Abstract: The main part of this thesis, Part II, consists of four papers. A summary and background is provided in Part I.Paper A introduces blow-ups of derived schemes in arbitrary centers, which is a generalization of the quasi-smooth case from [KR19]. The blow-up BlZX of a closed immersion j : Z → X of derived schemes is defined as the projective spectrum of the derived Rees algebra associated to j. The main result of Paper A concerns the existence of these Rees algebras, for which derived Weil restrictions are used. In order to make sense of this construction, Paper A generalizes the duality between Z-graded algebras and affine Gm -schemes, familiar from classical algebraic geometry, to the derived setting. The former are defined as derived algebras over a Lawvere-style theory, which produces an ∞-category of M-graded simplicial algebras, for any commutative monoid M.The main open question not answered in Paper A is, whether the description of derived blow-ups in quasi-smooth centers in terms of virtual Cartier divisors goes through in the general setting. This is answered affirmatively in Paper B, based on a detailed study of derived Weil restrictions. This includes an algebraicity result for Weil restrictions along affine morphisms of finite Tor-amplitude, which can be of independent interest.Paper B is also more general than Paper A, since it deals with blow-ups of closed immersions of derived stacks. The main construction is the derived deformation space via Weil restrictions, which leads to a deformation to the normal bundle for any morphism of derived stacks which admits a cotangent complex.The viewpoint from Paper B reveals that the central constructions are purely formal, so it is natural to ask for a further generalization. This is provided by Paper C. Here, blow-ups are defined in an axiomatic setting for nonconnective derived geometry—where the affine building blocks are the spectra of nonconnective LSym-algebras in a given derived algebraic context, in the sense of Bhatt–Mathew [Rak20]. Paper C first proposes a globalization based on these building blocks, and then develops the basic theory needed in order to carry out the blow-up construction and the deformation to the normal bundle in such a geometric context. The main example of this, besides the one for derived algebraic geometry, is derived analytic geometry. Paper C leads to a significantly more streamlined proof of the existence of the Rees algebra. This is because, in the nonconnective setting, the deformation space DX/Y is equivalent to the relative spectrum Spec RextX/Y of the nonconnective, extended Rees algebra of X → Y, for any affine morphism X → Y. Together with the algebraicity results from Paper B, this can provide an interesting test-case for understanding the relationship between nonconnectivity and algebraicity.The main application of derived blow-ups in this thesis, provided in Paper D, is a reduction of stabilizers algorithm for derived 1-algebraic stacks over C with good moduli spaces on their classical truncations. This is done using a derived Kirwan resolution, using derived intrinsic blow-ups—the classical versions of which are used in [Sav20, KLS17] for a reduction of stabilizers of classical Artin stacks. Paper D then proceeds with successive blow-ups of the derived locus of maximal stabilizer dimension. This is a generalization of the classical case defined in [ER21], where it is used for another reduction of stabilizers algorithm. The results of Paper D also explains the difference between these two approaches.