A bound on the degree of schemes defined by quadratic equations

We consider complex projective schemes $X\subset {\mathbb{P}}^r$ 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 $X\subset {\mathbb{P}}^r$ is bounded by a function of its codimension, $c$, whose asymptotic behaviour is given by $2^c\mathrm{/}\sqrt[4]{\pi c}$, thus improving the obvious bound $d\le 2^c$. More precisely, we get the bound $\left(\genfrac{}{}{0px}{}{d}{2}\right)\le \left(\genfrac{}{}{0px}{}{2c-1}{c-1}\right)$. Furthermore, if $X$ satisfies property $N_p$ or $N_{2,p}$ we obtain the better bound $\left(\genfrac{}{}{0px}{}{d+2-p}{2}\right)\le \left(\genfrac{}{}{0px}{}{2c+3-2p}{c+1-p}\right)$. Some classification results are also given when equality holds.