Suppose $P$ be a zero dimensional subscheme of length $m$ in $\mathbb{P}^3$ which is in linearly general position. Further assume that there is a degree $d$ form passing through $P$. Can $P$ satisfy Cayley-Bacharach property for $\mathcal{O}(d)$

I got the answer of the question. In fact it can satisfy Cayley-Bacharach. For example if we choose 8 points in rational cubic curve in linearly general position then it always satisfies C-B for $\mathcal{O}(2)$.

  • $\begingroup$ why if $P$ is contained a unique curve of degree $d$, then it satisfies Cayley-Bacharach property for $\mathcal{O}(d)$ ? Is it possible to give some lower bound on number of independent conditions imposed by $m$ points in linearly general position on sections of $\mathcal{O}(d)$ ? $\endgroup$ – user130022 May 21 at 12:20
  • $\begingroup$ We say $P$ satisfy CB for $\mathcal{O}(d)$ if any section of $\mathcal{O}(d)$ which vanishes at a co-length $1$ subscheme then it vanishes on $P$. $\endgroup$ – user130022 May 21 at 12:40
  • $\begingroup$ Ah OK, sorry, I had something different in mind. I delete my previous comments. $\endgroup$ – abx May 21 at 12:59
  • $\begingroup$ In any case can we give a a lower bound on the number of independent conditions imposed on sections of $\mathcal{O}(d)$ by $m$ points in general position ? $\endgroup$ – user130022 May 21 at 13:15
  • $\begingroup$ If $m\leq \frac{1}{2} (d+1)(d+2)=\dim H^0(\mathscr{O}_{\mathbb{P}}(d))$, $m$ general points impose $m$ independent conditions. $\endgroup$ – abx May 21 at 14:19

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