We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field
k
k
, this enrichment counts the number of lines meeting four lines defined over
k
k
in
P
k
3
\mathbf {P}^3_k
, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in
A
1
\mathbf {A}^1
-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms.
This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta function of the curves in this class over the field of definition of all automorphisms in the subgroup.
We prove a conductor-discriminant inequality for all hyperelliptic curves defined over discretely valued fields K with perfect residue field of characteristic not 2. Specifically, if such a curve is given by y, and if X is its minimal regular model over O K , then the negative of the Artin conductor of X (and thus also the number of irreducible components of the special fiber of X ) is bounded above by the valuation of disc(f ). There are no restrictions on genus of the curve or on the ramification of the splitting field of f . This generalizes earlier work of Ogg, Saito, Liu, and the second author.The proof relies on using so-called Mac Lane valuations to resolve singularities of arithmetic surfaces, a technique that was recently introduced by Wewers and the first author. Contents 1. Introduction Notation and conventions Acknowledgements 2. The discriminant bonus and regular models 3. A key reduction step 4. Mac Lane valuations 5. Mac Lane valuations, normal models and regular resolutions 6. Regular horizontal divisors 7. Discriminant bonus calculations 8. Comparing number of components and the discriminant bonus 9. Proof of the conductor-discriminant inequality in the irreducible case 10. Proof of the conductor-discriminant inequality in the reducible case Appendix A. Examples References
A field K is called ample if for every geometrically integral K-variety V with a smooth K-point, V (K) is Zariski-dense in V . A field K is virtually ample if some finite extension of K is ample. We prove that there exists a virtually ample field that is not ample.Date: October 15, 2018.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citationsācitations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.