A remark on Ulrich and ACM bundles

Research output: Contribution to journalArticle

Abstract

I show that on any smooth, projective ordinary curve of genus at least two and a projective embedding, there is a natural example of a stable Ulrich bundle for this embedding: namely the sheaf B X 1 of locally exact differentials twisted by O X (1) given by this embedding and in particular there exist ordinary varieties of any dimension which carry Ulrich bundles. In higher dimensions, assuming X is Frobenius split variety I show that B X 1 is an ACM bundle and if X is also a Calabi–Yau variety and p>2 then B X 1 is not a direct sum of line bundles. In particular I show that B X 1 is an ACM bundle on any ordinary Calabi–Yau variety. I also prove a characterization of projective varieties with trivial canonical bundle such that B X 1 is ACM (for some projective embedding datum): all such varieties are Frobenius split (with trivial canonical bundle).

Original languageEnglish (US)
Pages (from-to)20-29
Number of pages10
JournalJournal of Algebra
Volume527
DOIs
StatePublished - Jun 1 2019

Keywords

  • ACM bundles
  • Calabi–Yau variety
  • Frobenius split varieties
  • Ordinary varieties
  • Ulrich bundles

ASJC Scopus subject areas

  • Algebra and Number Theory

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