A criterion for virtual global generation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 39-53.

Let X be a smooth projective curve defined over an algebraically closed field k, and let F X denote the absolute Frobenius morphism of X when the characteristic of k is positive. A vector bundle over X is called virtually globally generated if its pull back, by some finite morphism to X from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of k is positive, a vector bundle E over X is virtually globally generated if and only if (F X m ) * EE a E f for some m, where E a is some ample vector bundle and E f is some finite vector bundle over X (either of E a and E f are allowed to be zero). If the characteristic of k is zero, a vector bundle E over X is virtually globally generated if and only if E is a direct sum of an ample vector bundle and a finite vector bundle over X (either of them are allowed to be zero).

Classification : 14H60, 14F05
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     title = {A criterion for virtual global generation},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
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Biswas, Indranil; Parameswaran, A. J. A criterion for virtual global generation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 39-53. http://www.numdam.org/item/ASNSP_2006_5_5_1_39_0/

[Ba] C. M. Barton, Tensor products of ample vector bundles in characteristic p, Amer. J. Math. 93 (1971), 429-438. | MR | Zbl

[Bi1] I. Biswas, Parabolic ample bundles, Math. Ann. 307 (1997), 511-529. | MR | Zbl

[Bi2] I. Biswas, A criterion for ample vector bundles over a curve in positive characteristic, Bull. Sci. Math. 129 (2005), 539-543. | MR | Zbl

[BP] I. Biswas and A. J. Parameswaran, On the ample vector bundles over curves in positive characteristic, C.R. Acad. Sci. Paris Sér. I Math. 339 (2004), 355-358. | MR | Zbl

[Br] H. Brenner, Slopes of vector bundles on projective curves and applications to tight closure problems, Trans. Amer. Math. Soc. 356 (2004), 371-392. | MR | Zbl

[Ha1] R. Hartshorne, “Ample Subvarieties of Algebraic Varieties”, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. | MR | Zbl

[Ha2] R. Hartshorne, Ample vector bundles on curves, Nagoya Math. J. 43 (1971), 73-89. | MR | Zbl

[La] H. Lange, On stable and ample vector bundles of rank 2 on curves, Math. Ann. 238 (1978), 193-202. | MR | Zbl

[Lan] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251-276. | MR | Zbl

[MS] V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205-239. | MR | Zbl

[Mu] D. Mumford, Pathologies III, Amer. J. Math. 89 (1967), 94-104. | MR | Zbl

[No] M. V. Nori, On the representations of the fundamental group scheme, Compositio Math. 33 (1976), 29-41. | Numdam | MR | Zbl

[Pa] A. J. Parameswaran, Virtual global generation of ample bundles over curves, unpublished manuscript.

[RR] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tôhoku Math. J. (2) 36 (1984), 269-291. | MR | Zbl