By Francis E. Burstall, John H. Rawnsley
During this monograph on twistor thought and its purposes to harmonic map thought, a vital subject matter is the interaction among the advanced homogeneous geometry of flag manifolds and the true homogeneous geometry of symmetric areas. particularly, flag manifolds are proven to come up as twistor areas of Riemannian symmetric areas. purposes of this conception comprise a whole type of strong harmonic 2-spheres in Riemannian symmetric areas and a B?cklund rework for harmonic 2-spheres in Lie teams which, in lots of instances, presents a factorisation theorem for such spheres in addition to hole phenomena. the most tools used are these of homogeneous geometry and Lie idea including a few algebraic geometry of Riemann surfaces. The paintings addresses differential geometers, specifically people with pursuits in minimum surfaces and homogeneous manifolds.
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Additional info for Twistor Theory for Riemannian Synmetric Spaces with Applications to Harmonic Maps of Riemann Surfaces
E6 ]. These divisors satisfy the intersection behaviour (Λ, Λ) = 1, (Λ, Ei ) = 0, (Ei , Ej ) = −1, if i = j, 0, if i = j. The remaining divisors may be expressed in terms of these elements via the relations [Ej ]. 3) [Li,j ] = [Λ] − [Ei ] − [Ej ], [Qi ] = 2[Λ] − j=i The adjunction formula implies that for any curve C ⊂ S of genus g, one has the relation (C, C + KS ) = 2g − 2. It easily follows that the class of the anticanonical divisor −KS is given by 6 [−KS ] = 3[Λ] − [Ej ]. i=1 6 One can check that the hyperplane section has class −3[Λ] + i=1 [Ej ] in PicQ (S), so that the cubic surface has very ample anticanonical divisor.
X5 ]. 18). By assumption S contains a conic deﬁned over Q. We may assume without loss of generality that the conic is contained in the plane x4 = x5 = 0. This means that there exists a ternary quadratic form Q ∈ Z[x1 , x2 , x3 ] such that Q | Qi (x1 , x2 , x3 , 0, 0) for i = 1, 2, whence Qi (x1 , x2 , x3 , 0, 0) = μi Q for certain μ1 , μ2 ∈ Z. We may therefore assume that S is deﬁned by the pair of quadratic forms Qi (x) = μi Q(x1 , x2 , x3 ) + Li (x1 , x2 , x3 )x4 + Mi (x1 , x2 , x3 )x5 + Pi (x4 , x5 ), where Li , Mi ∈ Q[x1 , x2 , x3 ] are linear and Pi ∈ Q[x4 , x5 ] is quadratic.
Let us begin with a discussion of non-singular del Pezzo surfaces. Let d 3. Then a del Pezzo surface of degree d is a non-singular surface S ⊂ Pd of degree d, with very ample anticanonical divisor −KS . This latter condition is equivalent to the equality [−KS ] = [H] in PicQ (S), for a hyperplane section H ∈ Div(S). The geometry of del Pezzo surfaces is very beautiful and well worth studying. However, to avoid straying from the main focus of this book, we will content ourselves with simply quoting the facts that are needed, referring the interested reader to the book by Manin .