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Birational transformations with isolated fundamental points

Published online by Cambridge University Press:  20 January 2009

J. A. Todd
J. A. Todd
Affiliation:
Manchester University
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It is well known that the canonical system of curves on an algebraic surface is only relatively invariant under birational transformations of the surface. That is, if we have a birational transformation T between two surfaces F and F′, and if K and K′ denote curves of the unreduced canonical systems on F and F′, then

where E and E′ denote the sets of curves, on F and F′ respectively which are transformed into the neighbourhoods of simple points on the other surface.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 5 , Issue 3 , February 1938 , pp. 117 - 124
Copyright
Copyright © Edinburgh Mathematical Society 1938

References

page 117 note 1 Todd, J. A., “The geometrical invariants of algebraic loci,” Proc. London Math. Soc. (2), 43 (1937), 127;Google Scholar “The arithmetical invariants of algebraic loci,” ibid. (2), 43 (1937), 190. We refer to these papers as I and II.

page 117 note 2 Segre, B., “Quelques resultats nouveaux dans la géométric sur une V 3 algébrique,” Mem. Acad. royale de Belgique (2), 14 (1936).Google Scholar

page 119 note 1 II , formula (4).

page 122 note 1 II, §7.

page 123 note 1 Segre, B., Mem. R. Ace. d'ltalia, 5 (1934), 505.Google Scholar

page 123 note 2 II, formula (37).

page 123 note 3 II, formula (56).

page 124 note 1 II, formula (61).