Rational points on symmetric powers and categorical representability

2017/04/08 by Saša Novaković, Novaković, Saša
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1704.02474

openalex publication_date 2017/04/08 · openalex created_date 2017/04/28 · openalex updated_date 2026/07/28

Abstract

In this paper we observe that for geometrically integral projective varieties X, admitting a full weak exceptional collection consisting of pure vector bundles, the existence of a k-rational point implies rdim(X)=0. We also study the symmetric power Sⁿ(X) of Brauer--Severi and involution varieties over ℝ and prove that the equivariant derived category D^b_Sn(Xⁿ) admits a full weak exceptional collection. As a consequence, we find rdim(X)=0 if and only if rdim(D^b_Sn(Xⁿ))=0 for 1≤ n≤ 3. If X is Brauer--Severi, the existence of a ℝ-rational point on X or S³(X) is equivalent to rdim(D^b_S3(X³))=0.

Citations

• On equivariant triangulated categories

Related

• Non-existence of exceptional collections on twisted flags and categorical representability via noncommutative motives
• On non-existence of full exceptional collections on some relative flags
• No phantoms in the derived category of curves over arbitrary fields, and derived characterizations of Brauer-Severi varieties
• Consequences of the existence of exceptional collections in arithmetic and rationality
• All strictly exceptional collections in D^b_coh(P^m) consist of vector bundles