Eisenbud–Green–Harris Cayley–Bacharach conjecture

Let $\Gamma$ be any subscheme of a zero-dimensional complete intersection of hypersurfaces of degrees $d_1\leq\dots\leq d_c$ in a projective space $P^c$. Say that $\Gamma$ fails to impose independent conditions on hypersurfaces of degree $m$ when the corresponding restriction conditions are not independent. Eisenbud–Green–Harris Cayley–Bacharach conjecture. If $\Gamma$ fails to impose independent conditions on hypersurfaces of degree $m$, then

$$\deg(\Gamma)\geq (e+1)d_{k+2}d_{k+3}\cdots d_c,$$

where $e$ and $k$ satisfy

$$\sum_{i=k+2}^c(d_i-1)\leq m+1<\sum_{i=k+1}^c(d_i-1)$$

and

$$e=m+1-\sum_{i=k+2}^c(d_i-1).$$

This is the Eisenbud–Green–Harris conjecture discussed in the Cayley–Bacharach section; the supplied text gives no resolution status for the full statement.