  The set of common roots of a finite set $I$ (it is an ideal) of homogeneous
polynomials is known as projective algebraic set $V$. In this article I show
how to dualize such projective algebraic sets $V$ by elimination of variables
from a system of polynomials with the Gr\"obner bases method. A dualization
algorithm is implemented in the computer algebra system {\sc Singular}. Some
examples are given. The main diagram shows the relationship between the ideal
$I$, its radical $\sqrt{I}$ and their dual ideals.
