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Vector bundles on proper toric 3-folds and certain other schemes. (English) Zbl 1430.14089

The main question of the paper is if every complete scheme \(X\) admits locally free sheaves \(\mathcal{E}\) of rank \(n=\dim X\) with \(c_n(\mathcal{E})\) arbitrarily large. This is related to similar questions like those asking for locally free sheaves of finite rank that are not free or for the existence of locally free resolutions of arbitrary coherent sheaves.
The authors obtain an affirmative answer for complete toric threefolds. However, as Payne has shown that there are complete toric threefolds such that all {\em toric} vector bundles of rank \(\leq 3\) are trivial, the sheaf \(\mathcal{E}\) cannot be toric in general. This answer is derived from the following (non-toric) result:
If \(X\to Y\) is a proper birational morphism with \(Y\) being complete, and if \(D\subset X\) is an effective Cartier divisor such that its associated scheme is projective and intersects the exceptional set in only finitely many points, then \(Y\) admits locally free sheaves \(\mathcal{E}\) of rank \(n=\dim Y\) with \(c_n(\mathcal{E})\) arbitrarily large.
While this assumption sounds quite technical, such a situation will be provided within a toric context. The combinatorial counter part of this situation is called “rays in Egyptian position”. This property is satisfied for all rays in dimension at most three, but it is a true condition in higher dimensions.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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[1] Abe, Takuro, The elementary transformation of vector bundles on regular schemes, Trans. Amer. Math. Soc., 359, 9, 4285-4295 (electronic) (2007) · Zbl 1128.14012
[2] Anderson, Dave; Payne, Sam, Operational \(K\)-theory, Doc. Math., 20, 357-399 (2015) · Zbl 1344.14010
[3] Artin, M., Algebraic approximation of structures over complete local rings, Inst. Hautes \'Etudes Sci. Publ. Math., 36, 23-58 (1969) · Zbl 0181.48802
[4] Atiyah, M. F., Vector bundles over an elliptic curve, Proc. London Math. Soc. (3), 7, 414-452 (1957) · Zbl 0084.17305
[5] Th\'eorie des intersections et th\'eor\`“eme de Riemann-Roch, S\'”eminaire de G\'eom\'etrie Alg\'ebrique du Bois-Marie 1966-1967 (SGA 6); Lecture Notes in Mathematics, Vol. 225, xii+700 pp. (1971), Springer-Verlag, Berlin-New York
[6] Bogomolov, F. A.; Landia, A. N., \(2\)-cocycles and Azumaya algebras under birational transformations of algebraic schemes, Algebraic geometry (Berlin, 1988), Compositio Math., 76, 1-2, 1-5 (1990) · Zbl 0729.14014
[7] Borel, Armand; Serre, Jean-Pierre, Le th\'eor\`eme de Riemann-Roch, Bull. Soc. Math. France, 86, 97-136 (1958) · Zbl 0091.33004
[8] Conrad, Brian; de Jong, A. J., Approximation of versal deformations, J. Algebra, 255, 2, 489-515 (2002) · Zbl 1087.14004
[9] Corti{\~n}as, G.; Haesemeyer, C.; Walker, Mark E.; Weibel, C., The \(K\)-theory of toric varieties, Trans. Amer. Math. Soc., 361, 6, 3325-3341 (2009) · Zbl 1170.19001
[10] Corti{\~n}as, G.; Haesemeyer, C.; Walker, Mark E.; Weibel, C., The \(K\)-theory of toric varieties in positive characteristic, J. Topol., 7, 1, 247-286 (2014) · Zbl 1309.14017
[11] Cox, David A.; Little, John B.; Schenck, Henry K., Toric varieties, Graduate Studies in Mathematics 124, xxiv+841 pp. (2011), American Mathematical Society, Providence, RI · Zbl 1223.14001
[12] Deligne, Pierre, Le th\'eor\`eme de plongement de Nagata, Kyoto J. Math., 50, 4, 661-670 (2010) · Zbl 1208.14012
[13] Edelsbrunner, Herbert, Algorithms in combinatorial geometry, EATCS Monographs on Theoretical Computer Science 10, xvi+423 pp. (1987), Springer-Verlag, Berlin · Zbl 0634.52001
[14] Edidin, Dan; Hassett, Brendan; Kresch, Andrew; Vistoli, Angelo, Brauer groups and quotient stacks, Amer. J. Math., 123, 4, 761-777 (2001) · Zbl 1036.14001
[15] Eagon, John A.; Hochster, M., \(R\)-sequences and indeterminates, Quart. J. Math. Oxford Ser. (2), 25, 61-71 (1974) · Zbl 0278.13008
[16] Eikelberg, Markus, The Picard group of a compact toric variety, Results Math., 22, 1-2, 509-527 (1992) · Zbl 0786.14031
[17] Ford, T. J.; Stimets, R., The Picard group of a general toric variety of dimension three, Comm. Algebra, 30, 12, 5771-5779 (2002) · Zbl 1056.14069
[18] Fujita, Takao, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30, 2, 353-378 (1983) · Zbl 0561.32012
[19] Fulton, William, Introduction to toric varieties, The William H. Roever Lectures in Geometry, Annals of Mathematics Studies 131, xii+157 pp. (1993), Princeton University Press, Princeton, NJ · Zbl 0813.14039
[20] Fulton, William; MacPherson, Robert, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc., 31, 243, vi+165 pp. (1981) · Zbl 0467.55005
[21] Gabber, O., On space filling curves and Albanese varieties, Geom. Funct. Anal., 11, 6, 1192-1200 (2001) · Zbl 1072.14513
[22] Gel{\cprime }fand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, x+523 pp. (1994), Birkh\"auser Boston, Inc., Boston, MA · Zbl 0827.14036
[23] Gharib, Saman; Karu, Kalle, Vector bundles on toric varieties, C. R. Math. Acad. Sci. Paris, 350, 3-4, 209-212 (2012) · Zbl 1244.14043
[24] Giraud, Jean, Cohomologie non ab\'elienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, ix+467 pp. (1971), Springer-Verlag, Berlin-New York · Zbl 0226.14011
[25] Goodman, Jacob Eli, Affine open subsets of algebraic varieties and ample divisors, Ann. of Math. (2), 89, 160-183 (1969) · Zbl 0159.50504
[26] Grivaux, Julien, Chern classes in Deligne cohomology for coherent analytic sheaves, Math. Ann., 347, 2, 249-284 (2010) · Zbl 1193.14026
[27] Gross, Philipp, The resolution property of algebraic surfaces, Compos. Math., 148, 1, 209-226 (2012) · Zbl 1242.14013
[28] Grothendieck, Alexander, Sur quelques points d’alg\`ebre homologique, T\^ohoku Math. J. (2), 9, 119-221 (1957) · Zbl 0118.26104
[29] Grothendieck, A.; Dieudonn{\'e}, J. A., El\'ements de g\'eom\'etrie alg\'ebrique. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 166, ix+466 pp. (1971), Springer-Verlag, Berlin · Zbl 0203.23301
[30] Grothendieck, A., \'El\'ements de g\'eom\'etrie alg\'ebrique. II. \'Etude globale \'el\'ementaire de quelques classes de morphismes, Inst. Hautes \'Etudes Sci. Publ. Math., 8, 222 pp. (1961)
[31] Grothendieck, A., \'El\'ements de g\'eom\'etrie alg\'ebrique. III. \'Etude cohomologique des faisceaux coh\'erents. I, Inst. Hautes \'Etudes Sci. Publ. Math., 11, 167 pp. (1961)
[32] Grothendieck, A., \'El\'ements de g\'eom\'etrie alg\'ebrique. IV. \'Etude locale des sch\'emas et des morphismes de sch\'emas. III, Inst. Hautes \'Etudes Sci. Publ. Math., 28, 255 pp. (1966) · Zbl 0144.19904
[33] Grothendieck, A., \'El\'ements de g\'eom\'etrie alg\'ebrique. IV. \'Etude locale des sch\'emas et des morphismes de sch\'emas IV, Inst. Hautes \'Etudes Sci. Publ. Math., 32, 361 pp. (1967) · Zbl 0153.22301
[34] Rev\^etements \'etales et groupe fondamental, S\'eminaire de G\'eom\'etrie Alg\'ebrique du Bois Marie 1960-1961 (SGA 1), augment\'e de deux expos\'es de M. Raynaud, Lecture Notes in Mathematics, Vol. 224, xxii+447 pp. (1971), Springer-Verlag, Berlin-New York
[35] Gubeladze, Joseph, Toric varieties with huge Grothendieck group, Adv. Math., 186, 1, 117-124 (2004) · Zbl 1061.14056
[36] Hartshorne, Robin, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, xiv+256 pp. (1970), Springer-Verlag, Berlin-New York · Zbl 0208.48901
[37] Kleiman, Steven L., Geometry on Grassmannians and applications to splitting bundles and smoothing cycles, Inst. Hautes \'Etudes Sci. Publ. Math., 36, 281-297 (1969) · Zbl 0208.48501
[38] Payne, Sam, Toric vector bundles, branched covers of fans, and the resolution property, J. Algebraic Geom., 18, 1, 1-36 (2009) · Zbl 1161.14039
[39] Kempf, G.; Knudsen, Finn Faye; Mumford, D.; Saint-Donat, B., Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, viii+209 pp. (1973), Springer-Verlag, Berlin-New York
[40] Kresch, Andrew, Flattening stratification and the stack of partial stabilizations of prestable curves, Bull. Lond. Math. Soc., 45, 1, 93-102 (2013) · Zbl 1266.14022
[41] Moishezon1969 B. Moishezon, The algebraic analog of compact complex spaces with a sufficiently large field of meromorphic functions. I, Math. USSR Izv. 3 (1969), 167-226.
[42] Oda, Tadao, Torus embeddings and applications, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 57, xi+175 pp. (1978), Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin-New York · Zbl 0417.14043
[43] Oda, Tadao, Convex bodies and algebraic geometry, Translated from the Japanese, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 15, viii+212 pp. (1988), Springer-Verlag, Berlin · Zbl 0628.52002
[44] Okonek, Christian; Schneider, Michael; Spindler, Heinz, Vector bundles on complex projective spaces, Progress in Mathematics 3, vii+389 pp. (1980), Birkh\"auser, Boston, Mass. · Zbl 0438.32016
[45] Poonen, Bjorn, Bertini theorems over finite fields, Ann. of Math. (2), 160, 3, 1099-1127 (2004) · Zbl 1084.14026
[46] Popescu, Dorin, General N\'eron desingularization, Nagoya Math. J., 100, 97-126 (1985) · Zbl 0561.14008
[47] Schr{\"o}er, Stefan, On non-projective normal surfaces, Manuscripta Math., 100, 3, 317-321 (1999) · Zbl 0987.14031
[48] Schr{\"o}er, Stefan; Vezzosi, Gabriele, Existence of vector bundles and global resolutions for singular surfaces, Compos. Math., 140, 3, 717-728 (2004) · Zbl 1060.14024
[49] Serre, J.-P., Modules projectifs et espaces fibr\'es \`“a fibre vectorielle. S\'”eminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Expos\'e 23, 18 pp. (1958), Secr\'etariat math\'ematique, Paris
[50] Swan, Richard G., N\'eron-Popescu desingularization. Algebra and geometry, Taipei, 1995, Lect. Algebra Geom. 2, 135-192 (1998), Int. Press, Cambridge, MA · Zbl 0954.13003
[51] Totaro, Burt, The resolution property for schemes and stacks, J. Reine Angew. Math., 577, 1-22 (2004) · Zbl 1077.14004
[52] Winkelmann, J{\"o}rg, Every compact complex manifold admits a holomorphic vector bundle, Rev. Roumaine Math. Pures Appl., 38, 7-8, 743-744 (1993) · Zbl 0813.32025
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