### The Plane Quintic With Five Cusps

American Journal of Mathematics, Vol. 49, No. 2 (April 1927), 197-214

Introduction

The quintic with five cusps (characterized by the Plücker numbers *m* = *n* = κ = *i* = 5; ν = τ = 0; *p* = 1) has been considered by del Pezzo, Field, and Basset. Field gives a general descriptive account of the appearance of the curve under various conditions on the coefficients in its equation; Basset mentions it as the limiting case under quintics with five nodes; but neither paper gives a detailed study of the curve. In such a study the del Pezzo work is fundamental. [Description of del Pezzo's results omitted.]

In the present investigation, which takes as starting point the del Pezzo work, it is proved that if five points in the plane are given as cusps on a quintic, the curve thus uniquely determined is unipartite, with the five cusps and five inflexions occurring alternately. Moreover, it is shown that from the five given points the cuspidal tangents, the inflexions, the inflexional tangents, as well as a series of ordinary points on the quintic, may be obtained by linear construction.

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