On the existence of loci with given singularities

Abstract

SOLOMON LEFSCHETZ Index §1.Introduction.23 2. Absolute invariants of a curve.23 3. Conditions for a double point.25 5.The maximum number of cusps.27 7. Discussion of the general problem.28 12. Existence of the curves.33 16.Existence of the curves of genus 1, 2,3,4 and 5. 37 20.Singularities of surfaces and higher varieties.39Introduction.1.The numerical relations existing between ordinary or so-called Plückerian singularities of a plane curve were determined as early as 1834 by Plücker, but the inverse question has been left almost untouched.It may be stated thus: To show the existence of a curve having assigned Plückerian characters; and is equivalent to the determination of the maximum of cusps km that a curve of order m and genus p may have.Veronese t has solved the question for rational curves.As an example of errors that have been committed in this direction, we cite Salmon-Fiedler's f statement that a seventhic may have 13 cusps, in which case however it is found that it would have ( -2 ) bitangents !In this paper a condition derived from the theory of invariants is given for the characteristics of a curve, a discussion of the a priori possible curves follows, and the existence of the curves within a certain range is then shown.Finally a few words will be said as to the extension of the theory to surfaces and higher varieties.Absolute invariants of a curve.2. Let «i, e%, • • •, es be the coefficients of a quantic Q in z\, x2, x3, and e'i,e't, •••,e'jt be the coefficients of Q',a quantic in (x[,x'2,x's) obtained from * Presented to the Society, September 12, 1911.

Locations

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