Rational maps of ℙn with prescribed fixed points and the smooth conic case

Abstract

We construct rational maps of ℙn which have a prescribed variety as a component of their fixed point set. The resulting maps fix a pencil of lines for the case of hypersurfaces; thus including the cases of plane curves. We also determine the Cremona maps among the constructed ones for quadratic hypersurfaces. Our methods are based on associated matrices of forms of constant degree and the "triple action" of G = PGL n+1 on them. We include a complete classification of these maps and matrices for the case of the smooth conic curve in ℙ2. We obtain invariants and canonical forms for the orbits of our matrices under the triple action of G, modulo syzygies of a row vector. We obtain invariants and canonical forms for the orbits of the constructed rational maps under conjugation by G.