Examples of surfaces which are Ulrich–wild

Casnati, Gianfranco

doi:10.1090/proc/14414

Examples of surfaces which are Ulrich–wild
HTML articles powered by AMS MathViewer

by Gianfranco Casnati

Proc. Amer. Math. Soc. 148 (2020), 5029-5043

DOI: https://doi.org/10.1090/proc/14414

Published electronically: September 24, 2020

PDF | Request permission

Abstract:

We give examples of surfaces which are Ulrich–wild, i.e., that support families of dimension $p$ of pairwise non–isomorphic, indecomposable, Ulrich bundles for arbitrarily large $p$.

References

Hirotachi Abo, Wolfram Decker, and Nobuo Sasakura, An elliptic conic bundle in $\mathbf P^4$ arising from a stable rank-$3$ vector bundle, Math. Z. 229 (1998), no. 4, 725–741. MR 1664785, DOI 10.1007/PL00004679
Marian Aprodu, Laura Costa, and Rosa Maria Miró-Roig, Ulrich bundles on ruled surfaces, J. Pure Appl. Algebra 222 (2018), no. 1, 131–138. MR 3680998, DOI 10.1016/j.jpaa.2017.03.007
Enrique Arrondo and Laura Costa, Vector bundles on Fano 3-folds without intermediate cohomology, Comm. Algebra 28 (2000), no. 8, 3899–3911. MR 1767596, DOI 10.1080/00927870008827064
Marian Aprodu, Gavril Farkas, and Angela Ortega, Minimal resolutions, Chow forms and Ulrich bundles on $K3$ surfaces, J. Reine Angew. Math. 730 (2017), 225–249. MR 3692019, DOI 10.1515/crelle-2014-0124
Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225, DOI 10.1007/978-3-642-57739-0
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, Ulrich bundles on abelian surfaces, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4609–4611. MR 3544513, DOI 10.1090/proc/13091
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
Jörgen Backelin and Jürgen Herzog, On Ulrich-modules over hypersurface rings, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 63–68. MR 1015513, DOI 10.1007/978-1-4612-3660-3_{4}
L. Borisov, A. Căldăraru, and A. Perry, Intersections of two grassmannianns in $\mathbb {P}^9$, Available at arXiv:1707.00534 [math.AG]; to appear in J. Reine Angew. Math..
Markus Brodmann and Peter Schenzel, Arithmetic properties of projective varieties of almost minimal degree, J. Algebraic Geom. 16 (2007), no. 2, 347–400. MR 2274517, DOI 10.1090/S1056-3911-06-00442-5
Marta Casanellas, Robin Hartshorne, Florian Geiss, and Frank-Olaf Schreyer, Stable Ulrich bundles, Internat. J. Math. 23 (2012), no. 8, 1250083, 50. MR 2949221, DOI 10.1142/S0129167X12500838
Gianfranco Casnati, Special Ulrich bundles on non-special surfaces with $p_g=q=0$, Internat. J. Math. 28 (2017), no. 8, 1750061, 18. MR 3681122, DOI 10.1142/S0129167X17500616
Gianfranco Casnati, Ulrich bundles on non-special surfaces with $p_g=0$ and $q=1$, Rev. Mat. Complut. 32 (2019), no. 2, 559–574. MR 3942927, DOI 10.1007/s13163-017-0248-z
Gianfranco Casnati, Daniele Faenzi, and Francesco Malaspina, Rank two aCM bundles on the del Pezzo threefold with Picard number 3, J. Algebra 429 (2015), 413–446. MR 3320630, DOI 10.1016/j.jalgebra.2015.02.008
Gianfranco Casnati, Daniele Faenzi, and Francesco Malaspina, Rank two aCM bundles on the del Pezzo fourfold of degree 6 and its general hyperplane section, J. Pure Appl. Algebra 222 (2018), no. 3, 585–609. MR 3710718, DOI 10.1016/j.jpaa.2017.04.021
Gianfranco Casnati, Matej Filip, and Francesco Malaspina, Rank two aCM bundles on the del Pezzo threefold of degree 7, Rev. Mat. Complut. 30 (2017), no. 1, 129–165. MR 3596030, DOI 10.1007/s13163-016-0213-2
Gianfranco Casnati and Federica Galluzzi, Stability of rank two Ulrich bundles on projective $K3$ surfaces, Math. Scand. 122 (2018), no. 2, 239–256. MR 3789442, DOI 10.7146/math.scand.a-101999
Emre Coskun, Rajesh S. Kulkarni, and Yusuf Mustopa, Pfaffian quartic surfaces and representations of Clifford algebras, Doc. Math. 17 (2012), 1003–1028. MR 3007683
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
O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France 110 (1982), no. 3, 319–346 (French, with English summary). With an appendix by A. Beauville. MR 688038
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
D. Faenzi, Ulrich sheaves on $K3$ surfaces, Available at arXiv:1807.07826 [math.AG].
D. Faenzi and J. Pons-Llopis, The CM representation type of projective varieties, Available at arXiv:1504.03819v2 [math.AG].
Francisco Javier Gallego and B. P. Purnaprajna, Normal presentation on elliptic ruled surfaces, J. Algebra 186 (1996), no. 2, 597–625. MR 1423277, DOI 10.1006/jabr.1996.0388
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
J. Herzog, B. Ulrich, and J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 (1991), no. 2-3, 187–202. MR 1117634, DOI 10.1016/0022-4049(91)90147-T
Yuko Homma, Projective normality and the defining equations of ample invertible sheaves on elliptic ruled surfaces with $e\geq 0$, Natur. Sci. Rep. Ochanomizu Univ. 31 (1980), no. 2, 61–73. MR 610593
Yuko Homma, Projective normality and the defining equations of an elliptic ruled surface with negative invariant, Natur. Sci. Rep. Ochanomizu Univ. 33 (1982), no. 1-2, 17–26. MR 703959
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
Paltin Ionescu, Embedded projective varieties of small invariants, Algebraic geometry, Bucharest 1982 (Bucharest, 1982) Lecture Notes in Math., vol. 1056, Springer, Berlin, 1984, pp. 142–186. MR 749942, DOI 10.1007/BFb0071773
Paltin Ionescu, Embedded projective varieties of small invariants. III, Algebraic geometry (L’Aquila, 1988) Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 138–154. MR 1040557, DOI 10.1007/BFb0083339
Yeongrak Kim, Ulrich bundles on blowing ups, C. R. Math. Acad. Sci. Paris 354 (2016), no. 12, 1215–1218 (English, with English and French summaries). MR 3573932, DOI 10.1016/j.crma.2016.10.022
Angelo Felice Lopez, Noether-Lefschetz theory and the Picard group of projective surfaces, Mem. Amer. Math. Soc. 89 (1991), no. 438, x+100. MR 1043786, DOI 10.1090/memo/0438
Rosa M. Miró-Roig and Joan Pons-Llopis, Representation type of rational ACM surfaces $X\subseteq \Bbb {P}^4$, Algebr. Represent. Theory 16 (2013), no. 4, 1135–1157. MR 3079796, DOI 10.1007/s10468-012-9349-z
Rosa M. Miró-Roig and Joan Pons-Llopis, $n$-dimensional Fano varieties of wild representation type, J. Pure Appl. Algebra 218 (2014), no. 10, 1867–1884. MR 3195414, DOI 10.1016/j.jpaa.2014.02.011
R. M. Miró–Roig and J. Pons–Llopis, Special Ulrich bundles on elliptic surfaces, Preprint.
D. G. Northcott, A first course of homological algebra, Cambridge University Press, Cambridge-New York, 1980. Reprint. MR 640096
Christian Okonek, Flächen vom Grad $8$ im $\textbf {P}^4$, Math. Z. 191 (1986), no. 2, 207–223 (German). MR 818665, DOI 10.1007/BF01164025
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
Joan Pons-Llopis and Fabio Tonini, ACM bundles on del Pezzo surfaces, Matematiche (Catania) 64 (2009), no. 2, 177–211. MR 2800010
K. Ranestad, On smooth surfaces of degree ten in the projective fourspace, thesis, Oslo, (1988).