Abstract
PROF. T. G. ROOM uses the symbol (│p,q│r, [n]) to denote the locus, in projection space [n] of n dimensions, whese equations are given by the vanishing of the determinants of order r + 1 of a matrix (xij), of p rows and q columns, whose elements are linear functions of the (general homogeneous) co-ordinates in [n]. This he calls a determinantal locus; particular determinantal loci are the conic in [2] and the quadric surface, cubic curve and cubic surface in [3], with respective symbols (│2,2│1, [2]), (│2,2│1, [3]), (│2,3│1,[3]) and (│3,3│2, [3]). (│p,q│r, [n]) may be projectively generated as the locus of the meets of the ∞ r(p-r) sets of corresponding [n — p + r r]'s of q projectively related [n—p]-stars, and, since (│p,q│r, [n]) is the same as (│q,p│r,[n]), there is a conjugate protective generation obtained by interchanging p and q. Thus the cubic curve in [3] may be regarded either as the locus of the meets of corresponding planes of three related pencils or as the locus of the meets of corresponding lines of two related point-stars. If p and q are equal, the two conjugate generations are of the same kind.
The Geometry of Determinantal Loci
By Prof. T. G. Room. Pp. xxviii + 483. (Cambridge: At the University Press, 1938.) 42s. net.
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B., D. The Geometry of Determinantal Loci. Nature 144, 960 (1939). https://doi.org/10.1038/144960a0
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DOI: https://doi.org/10.1038/144960a0