  Let $L$ be a very ample line bundle on a projective scheme $X$ defined over
an algebraically closed field $\Bbbk$ with ${\rm char}~\Bbbk \neq 2$. We say
that $(X,L)$ satisfies property $\mathsf{QR}(k)$ if the homogeneous ideal of
the linearly normal embedding $X \subset \mathbb{P}H^0 (X,L)$ can be generated
by quadrics of rank $\leq k$. Many classical varieties such as Segre-Veronese
embeddings, rational normal scrolls and curves of high degree satisfy property
$\mathsf{QR}(4)$.
  In this paper, we first prove that if ${\rm char}~\Bbbk \neq 3$ then
$(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (d))$ satisfies property
$\mathsf{QR}(3)$ for all $n \geq 1$ and $d \geq 2$. We also investigate an
asymptotic behavior of property $\mathsf{QR}(3)$ for any projective scheme.
Namely, we prove that $(i)$ if $X \subset \mathbb{P} H^0 (X,L)$ is $m$-regular
then $(X,L^d )$ satisfies property $\mathsf{QR}(3)$ for all $d \geq m$ and
$(ii)$ if $A$ is an ample line bundle on $X$ then $(X,A^d )$ satisfies property
$\mathsf{QR}(3)$ for all sufficiently large even number $d$. These results
provide an affirmative evidence for the expectation that property
$\mathsf{QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in
the cases of Green-Lazarsfeld's condition $\mathrm{N}_p$ and
Eisenbud-Koh-Stillman's determininantal presentation in [EKS88]. Finally, when
${\rm char}~\Bbbk = 3$ we prove that $(\mathbb{P}^n ,
\mathcal{O}_{\mathbb{P}^n} (2))$ fails to satisfy property $\mathsf{QR}(3)$ for
all $n \geq 3$.
