  We consider complex projective schemes $X\subset\Bbb{P}^{r}$ defined by
quadratic equations and satisfying a technical hypothesis on the fibres of the
rational map associated to the linear system of quadrics defining $X$. Our
assumption is related to the syzygies of the defining equations and, in
particular, it is weaker than properties $N_2$, $N_{2,2}$ and $K_2$. In this
setting, we show that the degree, $d$, of $X\subset\Bbb{P}^{r}$ is bounded by a
function of its codimension, $c$, whose asymptotic behaviour is given by
${2^c}/{\sqrt[4]{\pi c}}$, thus improving the obvious bound $d\leq 2^c$. More
precisely, we get the bound $\binom{d}{2}\leq\binom{2c-1}{c-1}$. Furthermore,
if $X$ satisfies property $N_p$ or $N_{2,p}$ we obtain the better bound
$\binom{d+2-p}{2}\leq\binom{2c+3-2p}{c+1-p}$. Some classification results are
also given when equality holds.
