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Ulrich complexity and categorical representability dimension

Saša Novaković

TL;DR

This work investigates Ulrich complexity $ ext{uc}((X,H))$ for Brauer–Severi varieties, twisted flags, and involution varieties, and connects it to the categorical representability dimension $ ext{rdim}(X)$. By exploiting base-change to splitting fields, period/index data, and semiorthogonal decompositions, the authors derive general bounds $ ext{uc}( ext{φ}_{pd}(X)) leq ext{ind}(X) ext{uc}(v_{pd}( ext{P}^n_E))$, prove concrete values in key non-split cases (e.g., non-split Brauer–Severi curves have $ ext{uc}( ext{φ}_{2d}(X))=2$) and show that the tempting equality $ ext{uc}( ext{φ}_p(X))= ext{rdim}(X)+1$ does not hold universally. They establish a criterion tying $ ext{uc}$ to $ ext{rdim}$ via Euler characteristics of Ulrich bundles on projective space, and discuss when equality can occur, especially in the period=index regime, while illustrating a broad spectrum of examples where the relation breaks or holds. The appendix outlines a Beilinson-type program to relate Ulrich complexity to $ ext{rdim}$ through endomorphism algebras in full weak exceptional collections, offering a general framework for arithmetic- and category-theoretic interactions in a broader setting.

Abstract

We investigate the Ulrich complexity of certain examples of Brauer--Severi varieties, twisted flags and involution varieties and establish lower and upper bounds. Furthermore, we relate Ulrich complexity to the categorical representability dimension of the respective varieties. We also state an idea why, in general, a relation between Ulrich complexity and categorical representability dimension may appear.

Ulrich complexity and categorical representability dimension

TL;DR

This work investigates Ulrich complexity for Brauer–Severi varieties, twisted flags, and involution varieties, and connects it to the categorical representability dimension . By exploiting base-change to splitting fields, period/index data, and semiorthogonal decompositions, the authors derive general bounds , prove concrete values in key non-split cases (e.g., non-split Brauer–Severi curves have ) and show that the tempting equality does not hold universally. They establish a criterion tying to via Euler characteristics of Ulrich bundles on projective space, and discuss when equality can occur, especially in the period=index regime, while illustrating a broad spectrum of examples where the relation breaks or holds. The appendix outlines a Beilinson-type program to relate Ulrich complexity to through endomorphism algebras in full weak exceptional collections, offering a general framework for arithmetic- and category-theoretic interactions in a broader setting.

Abstract

We investigate the Ulrich complexity of certain examples of Brauer--Severi varieties, twisted flags and involution varieties and establish lower and upper bounds. Furthermore, we relate Ulrich complexity to the categorical representability dimension of the respective varieties. We also state an idea why, in general, a relation between Ulrich complexity and categorical representability dimension may appear.

Paper Structure

This paper contains 8 sections, 29 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Brauer--Severi varieties
  3. Generalities on Ulrich bundles
  4. Weak exceptional collections and semiorthogonal decompositions
  5. Ulrich complexity of Brauer--Severi varieties and twisted flags
  6. A relation between Ulrich complexity and $\mathrm{rdim}$
  7. Further examples
  8. Appendix

Key Result

Theorem 1

Let $X$ be a Brauer--Severi variety of dimension $3$ where $\mathcal{O}_X(2)$ exists. Suppose there is a smooth geometrically connected genus one curve $C$ on $X$ such that the restriction map $H^0(X,\mathcal{O}_X(2))\rightarrow H^0(C,\mathcal{O}_C(2))$ is bijective. Then there is an unique Ulrich b

Theorems & Definitions (58)

  • Theorem : NV1, Theorem 5.15
  • Corollary
  • Theorem
  • Theorem 2.1: KANG, Theorem 1 or LIV, Corollary 3.6
  • Lemma 3.1: BV, (3.6)
  • Proposition 3.2
  • proof
  • Proposition 3.3: see BV, Corollary 3.2
  • Definition 4.1
  • Definition 4.2
  • ...and 48 more