Virtual Cartier divisors and blow-ups

TL;DR

The work extends the classical blow-up universal property to the derived setting via virtual effective Cartier divisors, proving a bijection between $X$-morphisms $S\to\tilde{X}$ and virtual Cartier divisors on $S$ over $(X,Z)$. It develops a robust theory of virtual and generalized Cartier divisors, establishing their equivalence with a derived stack presentation $[\,A^1/G_m]$ and enabling a derived blow-up Bl_Z X that is representable, base-change compatible, and carries a universal exceptional divisor. The construction applies to quasi-smooth closed immersions and yields a deformation to the normal cone in derived geometry, with tools for virtual fundamental classes and applications to descent in homotopy K-theory. The framework further generalizes to simultaneous blow-ups in multiple centers, preserving key properties (properness, quasi-smoothness, and covariance) and enabling étale descent, thereby broadening the scope of derived intersection theory and deformation techniques.

Abstract

We prove a universal property for blow-ups in regularly immersed subschemes, based on a notion we call "virtual effective Cartier divisor". We also construct blow-ups of quasi-smooth closed immersions in derived algebraic geometry.

Theorems & Definitions (37)

• Proposition 2.2.4
• Example 2.3.2
• Example 2.3.3
• Lemma 2.3.5
• proof
• Proposition 2.3.8
• proof
• Example 2.3.9
• Example 2.3.10
• Proposition 2.3.14