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

This project was partially supported by the B. von Beskows foundation, the CMSC of the University of Haifa, the G.S. Magnusons foundation, the Göran Gustafsson foundation, the Hierta–Retzius foundation, and the Signeul foundation.

QC 221202

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 Bl_ZX 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 G_m -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 D_X/Y is equivalent to the relative spectrum Spec R^ext_X/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.

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