On Homaloidal Polynomials

摘要

Let P~n be the projective space over a field k. If F is a homogeneous polynomial, we say that F is homaloidal if the polar map partial deriv F defined by the partial derivatives of F is a birational selfmap of P~n. Although the problem of determining homaloidal polynomials has a classical flavor, the theme was only recently raised in an algebro-geometric context by Dolgachev [Do] following suggestions stemming from the theory of prehomogeneous varieties: the relative invariants of prehomo-geneous spaces are, in fact, homaloidal polynomials [EKP; KiSa]. Dolgachev classifies square free homaloidal polynomials in P~2 (see also [D]) and characterizes square free homaloidal polynomials in P~3 that are products of four independent linear forms. Dolgachev also raises the following question: Is it true that a non-square free product of linear forms is homaloidal if and only if the product of its factors with multiplicity 1 is? This question has been given a positive answer in a specific case (see [KrS]) and in full generality (see [DP]) in a topological context. We will give an algebraic proof of the following result.