Caption: Mathematical Models by Arnold Emch - Series 4 (1928)
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C. VARIOUS ALGEBRAIC C U R V E S Lantern Slides for Screen-projections 41. Construction of Hyper elliptic Curves. It is well k n o w n that an hyperelliptic curve is characterized b y the existence of One g2. T h e joins of the couples (P, P') of this series envelope in general a curve K m of class m = n — p — 1, if n denotes the order of the hyperelliptic C n of genus p. Such a birational transformation (P ^=> P') of the C n into itself can be incorporated in an involutorial C r e m o n a transformation of the plane of the curve. Hence every hyperelliptic curve m a y be obtained as the locus of pairs of corresponding points on the tangents of rational curves in certain involutorial Cremona transformations.a Fig.18 A particularly simple construction for a special involutorial quadratic transformation is obtained b y choosing as the invariant quadrangle. aA paper by the author on the theory of such constructions will soon appear in the Tohoku Mathematical Journal. See also same Journal, vols. 21 (1922), pp. 310-326; 24 (1925), pp. 68-87; 25 (1925), pp. 63-76; also A m . J. of Math., vol. 48 (1926), pp. 21-44. 28
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