[...]... nd(BU ), = which is simply the local version of the central simple algebra equivalence Now we have the tools to give the denition of the Brauer group we will use Denition 7 The Brauer group of a surface classes of Azumaya algebras on identity is the class of OS , S over k S over a eld k is the set of equivalence The group operation is tensor product, the and the inverse of A is the opposite algebra,... toward nding these obstructions on a given surface is to calculate the Brauer group so that the set (see Denition 9) B can be determined As mentioned above, the Brauer groups of Del Pezzo surfaces may be accessible because of the concrete description of exceptional divisors and therefore the possibility of building Brauer group elements from divisors that satisfy the conditions of Lemma 2 Furthermore,... in [ One might wonder if such a correspondence could be proven for Del Pezzo surfaces of other degrees The following theorem describes possible torsion elements in the Brauer group of degree one Del Pezzo surfaces Theorem 8 Let S1 /k be a Del Pezzo surface of degree 1 Then H 1 (Gk , P ic (S1 )) is isomorphic to one of the following groups • (Z/2Z)i , i = 0, 1, 2, 3, 4, 5, 6, 7, 8 • (Z/3Z)i , i = 1,... (σx − x)} Proof Apply H i (G, ·) for σ∈G and [cx ] denotes the cohomology class of to the short exact sequence cx ×n 0 → P − P → P/nP → 0 → The isomorphism then follows from the denition of the connecting homomorphism In the case of degree three Del Pezzo surfaces, Swinnerton-Dyer gave a description of the three-torsion elements of the Brauer group In particular, he described Galois- theoretic conditions... For these reasons, we focus on nding 5-torsion elements of the Brauer group of Del Pezzo surfaces of degree one 3.2 Outline of the Method In this subsection we sketch the outline of our result and proof Our overall goal is to better understand the rational points of a degree one Del Pezzo surface, B is an approximation to set S1 (k), Since the set we may hope to accomplish our goal by determining the. .. there is a family of surfaces which suggests itself for our purposes, since there is a highly structured conguration of lines on each of the surfaces in the family This is the family of Del Pezzo surfaces 2.4 Del Pezzo Surfaces Del Pezzo surfaces admit a range of equivalent descriptions The following denition from [ Har77] will be the most convenient for our purposes Denition 11 degree n Let n be a natural... the following divisors, relative to the blowing up map: • the full transform of the points • the strict transform of the lines through two points these fteen lines by • pi Denote these six lines by pi and e1 , , e6 pj (i = j ) fij the strict transform of the conics through each of the points except lines are denoted by Denote pi These six gi 2.4.2 Degree One Del Pezzo Surfaces Similarly, there... Z/2Z • Z/6Z ⊕ Z/3Z Proof [ Cor05, p.23] This does not imply that all such groups are realized as the degree one Del Pezzo surface, The groups Z/5Z and (Z/5Z)2 S1 H 1 (Gk , P ic (S1 )) for some The above groups are simply the only possibilities are of interest because they have the largest prime-order 28 elements of the listed groups One might also hope that the calculations of the set simplied if... imply that then weak approximation is closed In this case there is said to be a Brauer- Manin obstruction to weak approximation Whether testing the Hasse principle or weak approximation, the set B is an approximation of To dene B, S(k) that is hopefully easier to determine than we need to recall the precise denition of the Brauer group S(k) itself 14 2.2 The Brauer Group S The Brauer group of that ∗... in order to nd the subgroups, G, which will give nontrivial ve-torsion Brauer group elements on degree one Del Pezzo surfaces We turn to this task in the next chapter 34 CHAPTER 4 Five Torsion in the Brauer Group of Degree One Del Pezzo Surfaces In this chapter we utilize the tools we have described above to compute for what groups G the quotient (P/5P )G P G /5P G is nontrivial Then we describe . Number: 3212825 3212825 2006 Copyright 2006 by Carter, Andrea C. UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17,