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A bound on the degree of schemes defined by quadratic equations

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 ${2^c}/{\sqrt[4]{πc}}$, 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.