## aCM sheaves on the double plane

HTML articles powered by AMS MathViewer

- 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

- Vincenzo Ancona and Giorgio Ottaviani,
*Some applications of Beilinson’s theorem to projective spaces and quadrics*, Forum Math.**3**(1991), no. 2, 157–176. MR**1092580**, DOI 10.1515/form.1991.3.157 - M. Atiyah,
*On the Krull-Schmidt theorem with application to sheaves*, Bull. Soc. Math. France**84**(1956), 307–317. MR**86358** - M. F. Atiyah,
*Vector bundles over an elliptic curve*, Proc. London Math. Soc. (3)**7**(1957), 414–452. MR**131423**, DOI 10.1112/plms/s3-7.1.414 - Edoardo Ballico, Sukmoon Huh, and Joan Pons-Llopis,
*aCM sheaves of pure rank two on reducible hyperquadrics*, J. Algebra**489**(2017), 73–90. MR**3686973**, DOI 10.1016/j.jalgebra.2017.06.025 - Arnaud Beauville,
*Determinantal hypersurfaces*, Michigan Math. J.**48**(2000), 39–64. Dedicated to William Fulton on the occasion of his 60th birthday. MR**1786479**, DOI 10.1307/mmj/1030132707 - Arnaud Beauville,
*An introduction to Ulrich bundles*, Eur. J. Math.**4**(2018), no. 1, 26–36. MR**3782216**, DOI 10.1007/s40879-017-0154-4 - R.-O. Buchweitz, G.-M. Greuel, and F.-O. Schreyer,
*Cohen-Macaulay modules on hypersurface singularities. II*, Invent. Math.**88**(1987), no. 1, 165–182. MR**877011**, DOI 10.1007/BF01405096 - Winfried Bruns and Jürgen Herzog,
*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956** - Marta Casanellas and Robin Hartshorne,
*Gorenstein biliaison and ACM sheaves*, J. Algebra**278**(2004), no. 1, 314–341. MR**2068080**, DOI 10.1016/j.jalgebra.2003.11.013 - Marta Casanellas and Robin Hartshorne,
*ACM bundles on cubic surfaces*, J. Eur. Math. Soc. (JEMS)**13**(2011), no. 3, 709–731. MR**2781930**, DOI 10.4171/JEMS/265 - A. J. Sommese, A. Biancofiore, and E. L. Livorni (eds.),
*Algebraic geometry*, Lecture Notes in Mathematics, vol. 1417, Springer-Verlag, Berlin, 1990. MR**1040546**, DOI 10.1007/BFb0083328 - Jinwon Choi, Kiryong Chung, and Mario Maican,
*Moduli of sheaves supported on quartic space curves*, Michigan Math. J.**65**(2016), no. 3, 637–671. MR**3542770**, DOI 10.1307/mmj/1472066152 - Laura Costa, Rosa M. Miró-Roig, and Joan Pons-Llopis,
*The representation type of Segre varieties*, Adv. Math.**230**(2012), no. 4-6, 1995–2013. MR**2927362**, DOI 10.1016/j.aim.2012.03.034 - L. Costa and R. M. Miró-Roig,
*$GL(V)$-invariant Ulrich bundles on Grassmannians*, Math. Ann.**361**(2015), no. 1-2, 443–457. MR**3302625**, DOI 10.1007/s00208-014-1076-9 - L. Costa and R. M. Miró-Roig,
*Homogeneous ACM bundles on a Grassmannian*, Adv. Math.**289**(2016), 95–113. MR**3439681**, DOI 10.1016/j.aim.2015.11.013 - Olivier Debarre and Laurent Manivel,
*Sur la variété des espaces linéaires contenus dans une intersection complète*, Math. Ann.**312**(1998), no. 3, 549–574 (French). MR**1654757**, DOI 10.1007/s002080050235 - David Eisenbud,
*Homological algebra on a complete intersection, with an application to group representations*, Trans. Amer. Math. Soc.**260**(1980), no. 1, 35–64. MR**570778**, DOI 10.1090/S0002-9947-1980-0570778-7 - David Eisenbud and Jürgen Herzog,
*The classification of homogeneous Cohen-Macaulay rings of finite representation type*, Math. Ann.**280**(1988), no. 2, 347–352. MR**929541**, DOI 10.1007/BF01456058 - David Eisenbud and Frank-Olaf Schreyer,
*Boij-Söderberg theory*, Combinatorial aspects of commutative algebra and algebraic geometry, Abel Symp., vol. 6, Springer, Berlin, 2011, pp. 35–48. MR**2810424**, DOI 10.1007/978-3-642-19492-4_{3} - David Eisenbud and Frank-Olaf Schreyer,
*Resultants and Chow forms via exterior syzygies*, J. Amer. Math. Soc.**16**(2003), no. 3, 537–579. With an appendix by Jerzy Weyman. MR**1969204**, DOI 10.1090/S0894-0347-03-00423-5 - Daniele Faenzi,
*Rank 2 arithmetically Cohen-Macaulay bundles on a nonsingular cubic surface*, J. Algebra**319**(2008), no. 1, 143–186. MR**2378065**, DOI 10.1016/j.jalgebra.2007.10.005 - Daniele Faenzi and Francesco Malaspina,
*Surfaces of minimal degree of tame representation type and mutations of Cohen-Macaulay modules*, Adv. Math.**310**(2017), 663–695. MR**3620696**, DOI 10.1016/j.aim.2017.02.007 - D. Faenzi and J. Pons-Llopis,
*The CM representation type of projective varieties*, preprint, arXiv:1504.03819 [math.AG]. - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**0463157** - Robin Hartshorne,
*Stable vector bundles of rank $2$ on $\textbf {P}^{3}$*, Math. Ann.**238**(1978), no. 3, 229–280. MR**514430**, DOI 10.1007/BF01420250 - Robin Hartshorne,
*Generalized divisors on Gorenstein schemes*, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 287–339. MR**1291023**, DOI 10.1007/BF00960866 - Daniel Huybrechts and Manfred Lehn,
*The geometry of moduli spaces of sheaves*, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR**2665168**, DOI 10.1017/CBO9780511711985 - M. M. Kapranov,
*On the derived categories of coherent sheaves on some homogeneous spaces*, Invent. Math.**92**(1988), no. 3, 479–508. MR**939472**, DOI 10.1007/BF01393744 - Horst Knörrer,
*Cohen-Macaulay modules on hypersurface singularities. I*, Invent. Math.**88**(1987), no. 1, 153–164. MR**877010**, DOI 10.1007/BF01405095 - N. Mohan Kumar, A. P. Rao, and G. V. Ravindra,
*Arithmetically Cohen-Macaulay bundles on hypersurfaces*, Comment. Math. Helv.**82**(2007), no. 4, 829–843. MR**2341841**, DOI 10.4171/CMH/111 - N. Mohan Kumar, A. P. Rao, and G. V. Ravindra,
*Arithmetically Cohen-Macaulay bundles on three dimensional hypersurfaces*, Int. Math. Res. Not. IMRN**8**(2007), Art. ID rnm025, 11. MR**2340104** - C. G. Madonna,
*ACM vector bundles on prime Fano threefolds and complete intersection Calabi-Yau threefolds*, Rev. Roumaine Math. Pures Appl.**47**(2002), no. 2, 211–222 (2003). MR**1979043** - C. Madonna,
*A splitting criterion for rank 2 vector bundles on hypersurfaces in $\textbf {P}^4$*, Rend. Sem. Mat. Univ. Politec. Torino**56**(1998), no. 2, 43–54 (2000). MR**1794445** - Christian Okonek, Michael Schneider, and Heinz Spindler,
*Vector bundles on complex projective spaces*, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2011. Corrected reprint of the 1988 edition; With an appendix by S. I. Gelfand. MR**2815674** - Giorgio Ottaviani,
*Spinor bundles on quadrics*, Trans. Amer. Math. Soc.**307**(1988), no. 1, 301–316. MR**936818**, DOI 10.1090/S0002-9947-1988-0936818-5 - Giorgio Ottaviani,
*Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics*, Ann. Mat. Pura Appl. (4)**155**(1989), 317–341. MR**1042842**, DOI 10.1007/BF01765948 - Jean-Pierre Serre,
*Faisceaux algébriques cohérents*, Ann. of Math. (2)**61**(1955), 197–278 (French). MR**68874**, DOI 10.2307/1969915 - Carlos T. Simpson,
*Moduli of representations of the fundamental group of a smooth projective variety. I*, Inst. Hautes Études Sci. Publ. Math.**79**(1994), 47–129. MR**1307297** - Amit Tripathi,
*Rank 3 arithmetically Cohen-Macaulay bundles over hypersurfaces*, J. Algebra**478**(2017), 1–11. MR**3621660**, DOI 10.1016/j.jalgebra.2017.01.014 - Charles A. Weibel,
*An introduction to homological algebra*, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR**1269324**, DOI 10.1017/CBO9781139644136

## 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