Important covariant curves and a complete system of invariants of the rational quartic curve

Abstract

It is well understood that different domains of rationality are useful ill discussing different properties of curves. Two domains are employed in the folSowing, to render possible the geometric interpretation of certain invariants and covariant loci of the rational plane quartic. Section 1 is divided into two parts: In the first part is given a straightfozward proof of the covariance of curves derived from Xn by a certain translation scheme; in the second part SALMONES work on the combinants of two binary quartics is applied to those covariant curves of the R4 which can be found as combinants. In Section 2 the most important invariants of the Rw are discussed, and four invariants are found in terms of which any other invariant relation on the R can be expressed algebraically; in this sense these four invariants constitute an algebraically complete system. Section 3 contains a treatment of the invariants of the 4 wllen the R4 is taken as the section of the Steiner Quartic Surface by a plane; in this schenle the invariants occur as symmetric functions of the coefficients of the cutting plane.