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A note on specializing semi-orthogonal decompositions

Weimufei Wu

TL;DR

This note provides a negative answer to the question of whether families of semi-orthogonal decompositions satisfy the existence part of the valuative criterion for properness. It constructs a smooth projective family of rational surfaces with a $K$-linear SOD that cannot be lifted to any $R$-linear SOD compatible with base change, leveraging deformation-invariance of $K_0$-groups and a phantom subcategory from Krah. The argument uses a blow-up of $\mathbb{P}^2$ at ten points to produce a phantom and then glues this data into a family over a DVR, deriving a contradiction from the non-vanishing of the phantom on general fibers. The result demonstrates obstructions to lifting SODs in families and informs the moduli and specialization behavior of derived-category decompositions in algebraic geometry.

Abstract

We prove that families of semi-orthogonal decompositions do not satisfy the existence part of the valuative criterion for properness, giving a negative answer to a question posed by Belmans, Okawa, and Ricolfi.

A note on specializing semi-orthogonal decompositions

TL;DR

This note provides a negative answer to the question of whether families of semi-orthogonal decompositions satisfy the existence part of the valuative criterion for properness. It constructs a smooth projective family of rational surfaces with a -linear SOD that cannot be lifted to any -linear SOD compatible with base change, leveraging deformation-invariance of -groups and a phantom subcategory from Krah. The argument uses a blow-up of at ten points to produce a phantom and then glues this data into a family over a DVR, deriving a contradiction from the non-vanishing of the phantom on general fibers. The result demonstrates obstructions to lifting SODs in families and informs the moduli and specialization behavior of derived-category decompositions in algebraic geometry.

Abstract

We prove that families of semi-orthogonal decompositions do not satisfy the existence part of the valuative criterion for properness, giving a negative answer to a question posed by Belmans, Okawa, and Ricolfi.
Paper Structure (4 sections, 8 theorems, 6 equations)

This paper contains 4 sections, 8 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Proof of the main result

Key Result

Theorem 1.1

There exists a smooth projective family of rational surfaces $X\rightarrow \mathrm{Spec}( R)$ ($R$ is a discrete valuation ring, and $K$ is its field of fractions) with a $K$-linear semi-orthogonal decomposition of $\mathrm {D^b}\mathrm{Coh}(X_K)$ denoted by $\langle \mathcal{A}_K, \mathcal{B}_K\ran

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 2.1: Belmans2020ModuliSO
  • Theorem 2.2: see Perry_2022
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.4: Krah, Theorem 1.1
  • proof : Proof of Theorem \ref{['1.1']}
  • Proposition 3.5
  • proof
  • ...and 3 more