Predegree Polynomials of Plane Configurations in Projective Space

Abstract

We work over an algebraically closed field of characteristic zero. The group PGL(4) acts naturally on PN which parameterizes surfaces of a given degree in P3. The orbit of a surface under this action is the image of a rational map PGL(4) ⊂ P15→PN. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general Pj, j being the dimension of the orbit. We find the predegrees and other invariants for all surfaces supported on unions of planes. The information is encoded in the so-called predegree polynomials , which possess nice multiplicative properties allowing us to compute the predegree (polynomials) of various special plane configurations. The predegree has both combinatorial and geometric significance. The results obtained in this paper would be a necessary step in the solution of the problem of computing predegrees for all surfaces.