Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T19:16:27.585Z Has data issue: false hasContentIssue false
  • English
  • Français

Exceptional Covers of Surfaces

Published online by Cambridge University Press:  20 November 2018

Jeffrey D. Achter
Jeffrey D. Achter*
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80523 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider a finite morphism $f\,:\,X\,\to \,Y$ of smooth, projective varieties over a finite field $\mathbb{F}$. Suppose $X$ is the vanishing locus in ${{\mathbb{P}}^{N}}$ of $r$ forms of degree at most $d$. We show that there is a constant $C$ depending only on $(N,\,r,\,d)$ and $\deg (f)$ such that if $\left| \mathbb{F} \right|\,>\,C$, then $f\,(\mathbb{F})\,:\,X(\mathbb{F})\,\to Y(\mathbb{F})$ is injective if and only if it is surjective.

Keywords

11G25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 3 , 01 September 2010 , pp. 385 - 393
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Cossec, F. R., Projective models of Enriques surfaces. Math. Ann. 265(1983), no. 3, 283–334. doi:10.1007/BF01456021Google Scholar
[2] Deligne, P., Cohomologie étale. In: Séminaire de Géométrie Algébrique du Bois-Marie SGA 4½, Lecture Notes in Mathematics 569, Springer-Verlag, New York, 1977.Google Scholar
[3] Deligne, P., La conjecture de Weil II. Inst. Hautes études Sci. Publ. Math. 52(1980), 137–252.Google Scholar
[4] Deligne, P. and Illusie, L., Relèvements modulo p 2 et décomposition du complexe de de Rham. Invent. Math. 89(1987), no. 2, 247–270. doi:10.1007/BF01389078Google Scholar
[5] Fried, M., On a theorem of MacCluer. Acta Arith. 25(1973/74), 121–126.Google Scholar
[6] Guralnick, R. M., Tucker, T. J., and Zieve, M. E., Exceptional covers and bijections on rational points. Int. Math. Res. No. 2007, no. 1, 20 pages.Google Scholar
[7] Kowalski, E., The large sieve, monodromy and zeta functions of curves. J. Reine Angew. Math. 601(2006), 29–69. doi:10.1515/CRELLE.2006.094Google Scholar
[8] Mumford, D., The canonical ring of an algebraic surface. Appendix to O. Zariski, The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface. Ann. Math. (2) 76(1962), 612–615, 1962. doi:10.2307/1970376Google Scholar
[9] Poonen, B., Bertini theorems over finite fields. Ann. of Math. (2) 160(2004), no. 3, 1099–1127. doi:10.4007/annals.2004.160.1099Google Scholar