  A symmetric planar central configuration of the Newtonian six-body problem
$x$ is called cross central configuration if there are precisely four bodies on
a symmetry line of $x$. We use complex algebraic geometry and Groebner basis
theory to prove that for a generic choice of positive real masses
$m_1,m_2,m_3,m_4,m_5=m_6$ there is a finite number of cross central
configurations. We also show one explicit example of a configuration in this
class. A part of our approach is based on relaxing the output of the Groebner
basis computations. This procedure allows us to obtain upper bounds for the
dimension of an algebraic variety. We get the same results considering cross
central configurations of the six-vortex problem.
