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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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aCM sheaves on the double plane
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by E. Ballico, S. Huh, F. Malaspina and J. Pons-Llopis PDF
Trans. Amer. Math. Soc. 372 (2019), 1783-1816 Request permission

Abstract:

The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a by-product, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on multiple hyperplanes and prove the existence of such bundles that do not split if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most $3/2$ on the double plane and describe the family of isomorphism classes of them.
References
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Additional Information
  • E. Ballico
  • Affiliation: Dipartimento di Matematica, Università di Trento, 38123 Povo (TN), Italy
  • MR Author ID: 30125
  • Email: edoardo.ballico@unitn.it
  • S. Huh
  • Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea
  • MR Author ID: 886034
  • Email: sukmoonh@skku.edu
  • F. Malaspina
  • Affiliation: Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
  • MR Author ID: 833101
  • Email: francesco.malaspina@polito.it
  • J. Pons-Llopis
  • Affiliation: Department of Engineering and Information Sciences and Mathematics, University of L’Aquila, Via Vetoio, Loc. Coppito I-67100 L’Aquila, Italy
  • Address at time of publication: Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
  • MR Author ID: 931485
  • ORCID: 0000-0001-5952-0279
  • Email: juan.ponsllopis@polito.it
  • Received by editor(s): December 20, 2016
  • Received by editor(s) in revised form: August 16, 2017, February 12, 2018, May 18, 2018, and May 28, 2018
  • Published electronically: April 12, 2019
  • Additional Notes: S. Huh is the corresponding author
    The first and third authors were partially supported by GNSAGA of INDAM (Italy) and MIUR PRIN 2015 \lq Geometria delle varietà algebriche\rq.
    The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2018R1C1A6004285 and No. 2016R1A5A1008055).
    The fourth author was supported by a Postdoctoral Fellowship, DISIM 2017-B0010.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 1783-1816
  • MSC (2010): Primary 14F05; Secondary 13C14, 16G60
  • DOI: https://doi.org/10.1090/tran/7627
  • MathSciNet review: 3976577