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Transactions of the American Mathematical Society

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aCM sheaves on the double plane


Authors: E. Ballico, S. Huh, F. Malaspina and J. Pons-Llopis
Journal: Trans. Amer. Math. Soc. 372 (2019), 1783-1816
MSC (2010): Primary 14F05; Secondary 13C14, 16G60
DOI: https://doi.org/10.1090/tran/7627
Published electronically: April 12, 2019
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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.


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Additional Information

E. Ballico
Affiliation: Dipartimento di Matematica, Università di Trento, 38123 Povo (TN), Italy
Email: edoardo.ballico@unitn.it

S. Huh
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea
Email: sukmoonh@skku.edu

F. Malaspina
Affiliation: Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
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
Email: juan.ponsllopis@polito.it

DOI: https://doi.org/10.1090/tran/7627
Keywords: Arithmetically Cohen-Macaulay sheaf, double plane, layered sheaf
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.
Article copyright: © Copyright 2019 American Mathematical Society