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Blow-ups and normal bundles in nonconnective derived geometry - Jeroen Hekking

A current development in derived algebraic geometry is the theory of derived rings (developed by Antiau, Bhatt, Mathew, Raksit, et al), which are nonconnective algebras that recover animated rings after restricting to the connective part. It turns out that this is a remarkably convenient framework for constructing derived Rees algebras. In this talk, we will review the theory of derived rings, and explain how it relates to animated rings and commutative ring spectra. We will then look at a way to geometrize the theory to nonconnective derived stacks. This will be the framework in which we define derived blow-ups in terms of derived Rees algebras, and where we exhibit a derived deformation to the normal bundle. The theory of derived rings is actually an axiomatic system that covers more than nonconnective derived algebraic geometry. A salient example is derived analytic geometry (in terms of Banach algebras), which we can review if time permits. A main application of derived blow-ups is a derived reduction of stabilizers algorithm, leading to sheaf-theoretic counting invariants. This is based on joint work with Khan–Rydh, with Ben-Bassat, and with Rydh–Savvas.