Logic and invariant theory II: Homogeneous coordinates, the introduction of higher quantities, and structural geometry

Abstract

In our previous paper [18], the interest of classical invariant theory in the algebraic form of projective geometric properties was recast in terms of invariance of first-order formulas for certain categories of models. Following the lead of the classical studies we concentrated on the nonsingular linear and semilinear transformations between vector spaces-the geometric collineations. We characterized, by their syntactic form, the first-order formulas invariant for these categories. Classical invariant theory has developed, since 1900, in algebraic rather than geometric hands and been pronounced dead by algebraists [6]. Perhaps for this reason the further question of invariance under choices of homogeneous coordinates was not explicitly developed. There appear to be several contributing factors. For technical convenience, polynomial concomitants were simply reduced to their homogeneous parts and these parts analyzed separately, without resorting to questions of invariance [4, p. 71. It happened that ?z-ary forms, which are already homogeneous, were the algebraically interesting quantities and the tools available handled these forms, at least for linear transformations [4, pp. 4-7; 13, p. 14; 14, pp. 133181. All q ucstions of invariance were placed in a rigid mold of inv-ariance under a group of transformations, and changing homogeneous coordinates at a particular variable did not produce any obvious group. IVe note that these changes of coordinates also fail to fit the framework of “invariance for a category of models” presented in [18]. These tendencies went along with the algebraic assumption that what is interesting in invariance is to have: