# Download Combinatorial Geometry in the Plane (translation of by Hugo Hadwiger, Hans Debrunner (translated by V. Klee) PDF

By Hugo Hadwiger, Hans Debrunner (translated by V. Klee)

Read Online or Download Combinatorial Geometry in the Plane (translation of Kombinatorische Geometrie in der Ebene, with a new chapter supplied by the translator) PDF

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Extra info for Combinatorial Geometry in the Plane (translation of Kombinatorische Geometrie in der Ebene, with a new chapter supplied by the translator)

Example text

This shows that the orbit V ( \ ) ' z is G^-invariant. Let us take a suitable identification of V4' with the affine space C" so that the point z is identified with the origin. 3 the group G^ acts on V4' linearly and has the point z in the closure of any orbit. So the action of G^ on V4' is a good C*-action. , TJ. We assume that each coordinate function T, is homogeneous of some degree q^> 0. Let R = \]{\)'z. Being an orbit of a unipotent group acting on an affine variety, R is closed in V4". Since it is G^-invariant, it can be given by a system of equations F^ == ...

Then the radical of the stabilizer G^ is of dimension k or less. Proof. 18. 3. Theorem. — Let B be a Borel subgroup ofG and let G act on X x G/B diagonally. Then Pic°(X x G/B) is abundant. In particular, when G is a torus, Pic^X) is abundant. Proof. — Notice first that Pic°(G/B) ^ ^(T), where T is a maximal torus contained in B (see [KKV], p. 65). We want to show that for any point (^^[B]) with reductive stabilizer the isotropy representation homomorphism P(^[B]) ^ Pic^X x G/B) ->^(G^,^])) is surjective.

H, where {HJ,gi are the walls that contain / and { H , } , g j are the walls that do not contain /. ) Since the set of walls is finite, we see that the set of cells is finite. Also it is obvious that any wall is a union of cells. 10. Lemma. — Let F be a cell. For any /, /' e F one has X^/) = X^/'). Denote this set by X^F). Then X^F') C X^F) if F n F' + 0, F + F'. Proof. — We already know that X^) === X^/'). 8 (iii)). This implies that X8^) = X8^) and hence X^) = X^'). Now,_if ^eX^F'), then M\x) ^ 0 for any / e F'.