Let $C$ be a non hyperelliptic complex algebraic curve of genus $g$, then the vector space $I_2(C)$ of quadrics containing the canonical image of $C$ is $\binom{g+1}{2}-h^0(2\omega_C) = (g-2)(g-3)/2$ dimensional. Moreover, if $g > 4$ then $C$ is the intersection of the nulls of all these quadrics (see ACGH VI.4.1 for a proof, I'm not sure how "classical" this is, or who originally proved it).
Question (edited following a comment from David Speyer) what is the least $d$ so that if $V\subset I_2(C)$ is any $d$ dimensional vector space, and $X$ is the intersection of the nulls of the quadrics in $V$, then the only irreducible component of $X$ which linearly spans $|\omega_C|^*$ is the canonical image of $C$ ?
I don't even know the generic bound, or indeed what is the bound for hyperelliptic curves (in which case the canonical curve is a rational normal curve).
