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**Additional resources for An Apparent Dependence of the Apex and Velocity of Solar Motion, as Determined from Radial Velocitie**

**Sample text**

Consider the C∗ action on C4 deﬁned by λ : (X1 , X2 , X3 , X4 ) → (λw1 X1 , λw2 X2 , λw3 X3 , λw4 X4 ) (diﬀerent combinatorial data). We deﬁne P3(w1 ,w2 ,w3 ,w4 ) = C4 \ {0} /C∗ . Suppose w1 = 1. Then choose λ = 1 such that λw1 = 1. 1 Since this singularity appears in codimension 3, a subvariety of codimension 1 will generically not intersect it — so it may not cause any problems. However, suppose (w2 , w3 ) = 1, so that k|w2 and k|w3 , with k > 1. Then choose λ = 1 such that λk = 1. Note (0, X2 , X3 , 0) = (0, λw2 X2 , λw3 X3 , 0), and we have a Z/kZ quotient singularity along a locus of points of codimension 2.

Then the global sections can be generated by the monomials X0n , X0n−1 X1 , . . , X1n . In short, the global sections are homogeneous polynomials of degree n. The same is true on PN : H 0 OÈN (n) = homogeneous polynomials of N +n−1 . In particular, degree n in X0 , . . , XN . So dim H 0 (O(n)) = n−1 the sections of OÈ4 (5) are quintic polynomials in ﬁve variables, and there are 9 · 8 · 7 · 6/4! = 126 independent ones. 2. The Cech–de Rham Isomorphism. (These few paragraphs are merely a summary of the treatment in [121], pp.

N, i as a basis for tangent vectors. We write ∇i X for dx , ∇X , the ith covector component of ∇X. Deﬁne Γ by ∇∂i ∂j = Γk ij ∂k . Then the torsion-free condition says Γk ij = Γk ji . Let us denote X, Y = g(X, Y ). 2. Start with ∂i gjk = ∂i ∂j , ∂k = ∇∂i ∂j , ∂k + ∂j , ∇∂i ∂k = m m m ij ∂m , ∂k + ∂j , Γ ik ∂m = Γij gmk +Γ ik gjm . Now add the equation with i ↔ j and subtract the equation with i ↔ k. Using the torsion-free condition Γk ij = Γk ji , show that Γm 1 Γi jk = g im (∂j gmk + ∂k gjm − ∂m gjk ).