Summary
We give an example of a vector bundle ε on a relative curve C → Spec ℤ such that the restriction to the generic fiber in characteristic zero is semistable but such that the restriction to positive characteristic p is not strongly semistable for infinitely many prime numbers p. Moreover, under the hypothesis that there exist infinitely many Sophie Germain primes, there are also examples such that the density of primes with nonstrongly semistable reduction is arbitrarily close to one.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Brenner, Computing the tight closure in dimension two, Math. Comput., 74–251 (2005), 1495–1518.
H. Brenner, There is no Bogomolov type restriction theorem for strong semistability in positive characteristic, Proc. Amer. Math. Soc., 133 (2005), 1941–1947.
D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Vieweg, Braunschweig, Germany, 1997.
K.-H. Indlekofer and A. Járai, Largest known twin primes and Sophie Germain primes, Mathematics of Computation, 68–227 (1999), 1317–1324.
Y. Miyaoka, The Chern class and Kodaira dimension of a minimal variety, in Algebraic Geometry, Sendai, 1985, Advanced Studies in Pure Mathematics 10, North-Holland, Amsterdam, 1987, 449–476.
J. P. Serre, Cours d’arithmétique, Presses Universitaires de France, Paris, 1970.
N. I. Shepherd-Barron, Semi-stability and reduction mod p, Topology, 37-3 (1997), 659–664.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Birkhäuser Boston
About this chapter
Cite this chapter
Brenner, H. (2005). On a Problem of Miyaoka. In: van der Geer, G., Moonen, B., Schoof, R. (eds) Number Fields and Function Fields—Two Parallel Worlds. Progress in Mathematics, vol 239. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4447-4_3
Download citation
DOI: https://doi.org/10.1007/0-8176-4447-4_3
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4397-3
Online ISBN: 978-0-8176-4447-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)