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CHQUASIGROUPS AND MOUFANG LOOPS
Symmetric Abelian quasigroups
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Abelian group automorphisms Azumaya algebra base field birational map birational morphism birationally trivial Brauer group bx(f calculation canonical representative CH-quasigroup Chapter closed point cocycle coincides commutative components composition law concludes the proof conjugate construction Corollary corresponding cubic curve cubic hypersurface cubic surface defined definition denote dense divisor elements embedding equal equation equivalence relation Example exceptional curves exists exponent fact fc-points fibre finite number formula G-modules Galois geometric graph group G Hasse principle homomorphism hyperplane induces intersection number invariant inverse image invertible sheaf irreducible isomorphism kernel linear lines minimal monoidal transformation morphism Moufang loops multiplicity non-empty non-singular non-trivial number field obtain P1 X P1 pair Pezzo surface Picard group plane proof of Theorem properties Proposition quadratic quasigroup R-equivalence rational map rational surfaces result sheaf space Spec subset symmetric quasigroup tangent tion verify VT(k W(Rr zero