I guess I am not getting something, as it seems elementary the maximum number of quadrics needed is usually $g$. I.e. given a canonical curve $C$ in $P^{g-1}$ which can be cut out by quadrics, it seems any general choice of $g-2$ quadrics containing it cuts out a union of curves including $C$. Then any general quadric containing $C$ cuts out $C$ and a finite set of points on the other curves. Then another general quadric through $C$ omits those points. Is this nonsense? I see now the question in the title is no longer the same as the edited question.

**edit:**
David, do you really want the property in your question or do you just want to know when every $d$ dimensional subspace of $I_2(C)$ determines the canonical curve somehow? i.e. a Torelli result.

Here is an example suggesting d may be large, a plane sextic, re embedded canonically in $P^9$ via plane cubics. The image is a del Pezzo surface $S$ of degree $9$, on which any one quadric cuts out the canonical curve, unless the quadric contains the del Pezzo. But the $55$ dimensional space of quadrics in $P^9$ cuts out the $28$ diml space of plane sextics, hence a $27$ diml space of quadrics contains the del Pezzo. Since the ideal $I_2$ has dimension $28$, we actually need the whole space $I_2$ to get the curve, or to get any set with the curve as a component.

Is this right? If so, plane curves of other degrees may be problematic as well....The situation seems to improve as the degree goes up. A plane septic seems to lie canonically on an embedded copy of $P^2$ that is contained in only $75$ independent quadrics among the $78$ containing the curve itself, so d seems to equal at least $76$, maybe $77$ since it seems to need two more quadrics this time. For a plane octic $d$ seems to be at least $166$, out of a space $I_2$ of dim = $171$. well we're gaining on it, but somewhat slowly.

1more comment