# Cayley-Bacharach property of points is linearly general position

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 $$\mathcal{O}(2)$$.

• 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 $\mathcal{O}(d)$ ? – user130022 May 21 at 12:20
• 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 $P$. – user130022 May 21 at 12:40
• Ah OK, sorry, I had something different in mind. I delete my previous comments. – abx May 21 at 12:59
• 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 ? – user130022 May 21 at 13:15
• 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. – abx May 21 at 14:19