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By Frances Clare Kirwan

These notes describe a basic technique for calculating the Betti numbers of the projective quotient types that geometric invariant idea affiliates to reductive team activities on nonsingular complicated projective forms. those quotient forms are fascinating specifically as a result of their relevance to moduli difficulties in algebraic geometry. the writer describes various methods to the matter. One is solely algebraic, whereas the opposite makes use of the tools of symplectic geometry and Morse idea, and comprises extending classical Morse thought to sure degenerate functions.

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Check that the Schwarzian derivative ( f )−2 [ f f −(3/2)( f )2 ] commutes with the action of (P1 ) = P S L(2, C) on the independent variable; see Ford [1972: 98–101] for more information and for applications of this intriguing object. 6 Spherical Geometry The sphere inherits from the ambient space R3 its customary (round) geometry, and it is easy to see that the shortest path (geodesic) joining two points is an arc of a great circle. Introduce spherical polar coordinates: x1 = sin ϕ cos θ, x2 = sin ϕ sin θ, x3 = cos ϕ in which 0 ≤ ϕ ≤ π is the colatitude measured from the north pole and 0 ≤ θ < 2π is longitude.

So h −2+2/d ≤ h/2, and this is contradictory unless h ≤ 3, that is, h = 2 or 3, h = 1 being the case d = 1, which is trivial. ) Case 1. h = 2. There are two classes and [m(k1 )]−1 + [m(k2 )]−1 = 2/d. Now m ≤ d divides d, and if m < d, then it is ≤ d/2 already and the identity cannot balance. In short, m = d for both classes and each comprises just one point. 2. Platonic solids Solid tetrahedron cube Faces Edges Corners m 4 triangles 6 4 3 6 squares 12 8 3 12 pentagons 30 20 5 octahedron 8 triangles 12 6 4 icosahedron 20 triangles 30 12 3 dodecahedron Case 2.

This can be cured. The branches are distinguished at x = 0 by their slopes y = ±1, which suggests a new attitude toward X, described afresh as the common roots (in P3 ) of the two relations x z = y and z 2 = 1 + x; in fact, z = y/x takes distinct values ±1 at x = 0, depending on the branch, so the old point (0, 0) splits into two separate points (0, 0 ± 1). The present discussion is just to whet the appetite; it is continued in Section 11, and also in Section 13. For further information, Bliss [1933], Walker [1978], Kirwan [1992], Shafarevich [1977], and Mumford [1976] are recommended, in order of sophistication.

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