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The Cremona Transformations of a Certain Plane Sextic
Author(s) -
Hudson Hilda P.
Publication year - 1917
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s2-15.1.385
Subject(s) - citation , plane (geometry) , mathematics , information retrieval , combinatorics , computer science , arithmetic , library science , geometry
A plane Cremona transformation of degree n transforms the straight lines of the first plane into a homaloidal family in the second plane ; that is, into a doubly infinite family of rational curves of degree n, such that two general curves of the family meet in one and only one variable point of intersection. The remaining intersections are all accounted for by the base points common to the whole family. The straight lines of the second plane are transformed into a homaloidal family in the first plane, also with a set of base points. To every point p in the first plane there corresponds a unique point P in the second plane : except that to a base point a there corresponds not a single point but a curve J. The set of curves J corresponding to the whole set of base points, taken with proper multiplicities, constitute the Jacobian of the second homaloidal family. To a straight line I passing through a there corresponds a homaloid which breaks up into J, corresponding to the single point a, and a rational curve 0 of degree less than n, corresponding to all the rest of I. Then J meets
corresponds properly to I. Each of Jx, J2 meets 0 in one point other than base points, and