  We prove the conjecture of Falikman--Friedland--Loewy on the parity of the
degrees of projective varieties of $n\times n$ complex symmetric matrices of
rank at most $k$.
  We also characterize the parity of the degrees of projective varieties of
$n\times n$ complex skew symmetric matrices of rank at most $2p$.
  We give recursive relations which determine the parity of the degrees of
projective varieties of $m\times n$ complex matrices of rank at most $k$.
  In the case the degrees of these varieties are odd, we characterize the
minimal dimensions of subspaces of $n\times n$ skew symmetric real matrices and
of $m\times n$ real matrices containing a nonzero matrix of rank at most $k$.
  The parity questions studied here are also of combinatorial interest since
they concern the parity of the number of plane partitions contained in a given
box, on the one hand, and the parity of the number of symplectic tableaux of
rectangular shape, on the other hand.
