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By Janos Kollar

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Additional resources for Complex Algebraic Geometry (Ias Park City Mathematics Series, V. 3) PCMS 3

Example text

Later requirements come in gradually. 1 Set notation For set notation we refer to Smith [12, pages 1 – 38]. We mention that we use the word function where it uses map. We should also like to emphasise the difference between a set {a, b} and a couple or ordered pair (a, b). In a set, the order of the elements does not matter, so that {a, b} = {b, a} in all cases, and {a, b} = {c, d} if and only if either a=c and b=d a=d and b = c. or In a couple (a, b) it matters which is first and which is second.

We have already seen that X ∈ [A, B] and X = B; as |A, X| > 0 we also have X = A. Uniqueness . Suppose now that Y ∈ l and |A, Y| = |Y, B|. g. e. 0 = |A, B|. 4 one of Y ∈ [A, B], A ∈ [Y, B], B ∈ [A,Y ], + + holds. The second of these would imply |Y, A| + |A, B| = |Y, B| and so |Y, A| < |Y, B| as |A, B| > 0. The third of these would imply |A, B| + |B, Y| = |A, Y| and so |B, Y| < |A, Y|. As these contradict our assumptions, we must have Y ∈ [A, B]. Then |A, Y| + |Y, B| = |A, B| and as |A, Y| = |Y, B| this implies that |A, Y| = 21 |A, B|.

By observation, we note that if A, B,C are non-collinear and [A, D is between [A, B and [A,C , then |∠BAD|◦ + |∠CAD|◦ = |∠BAC|◦ , while if [A, B and [A,C are opposite and D ∈ AB, then |∠BAD|◦ + |∠CAD|◦ = 180. By observation, given any number k with 0 ≤ k < 180 and any half-line [A, B , on each side of the line AB there is a unique wedge-angle ∠BAC with |∠BAC|◦ = k. In all cases |∠BAB|◦ = 0, so that the degree-measure of each null angle is 0, while if ∠BAC is not null then |∠BAC|◦ > 0. It follows from the foregoing, that if ∠BAD is any wedge-angle then |∠BAD|◦ < 180, and that if ∠BAD, ∠CAD are supplementary angles, then |∠CAD|◦ = 180 − |∠BAD|◦ .

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