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I guess I am not getting something, as it seems elementary the maximum number of quadrics needed is usually g. $g$. I.e. given a canonical curve C $C$ in P^(g-1) $P^{g-1}$ which can be cut out by quadrics, it seems any general choice of g-2 $g-2$ quadrics containing it cuts out a union of curves including C. $C$. Then any general quadric containing C $C$ cuts out C $C$ and a finite set of points on the other curves. Then another general quadric through C $C$ 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 $d$ dimensional subspace of I2(C) $I_2(C)$ 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 $P^9$ via plane cubics. The image is a del Pezzo surface S $S$ of degree 9, $9$, on which any one quadric cuts out the canonical curve, unless the quadric contains the del Pezzo. But the 55 $55$ dimensional space of quadrics in P^9 $P^9$ cuts out the 28 $28$ diml space of plane sextics, hence a 27 $27$ diml space of quadrics contains the del Pezzo. Since the ideal I2 $I_2$ has dimension 28, $28$, we actually need the whole space I2 $I_2$ 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 $P^2$ that is contained in only 75 $75$ independent quadrics among the 78 $78$ containing the curve itself, so d seems to equal at least 76, $76$, maybe 77 $77$ since it seems to need two more quadrics this time. For a plane octic d $d$ seems to be at least 166, $166$, out of a space I2 $I_2$ of dim = 171. $171$. well we're gaining on it, but somewhat slowly.

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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 I2(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 I2 has dimension 28, we actually need the whole space I2 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 I2 of dim = 171. well we're gaining on it, but somewhat slowly.

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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 I2(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 I2 has dimension 28, we actually need the whole space I2 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. For a plane octic d seems to be at least 166, out of a space I2 of dim = 171. well we're gaining on it.

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