On the Eisenbud-Green-Harris Conjecture

It has been conjectured by Eisenbud, Green and Harris that if $I$ is a homogeneous ideal in $k[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ containing a regular sequence $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n$ of degrees $\mathrm{deg}(f_i)=a_i$, where $2\le a_1\le \mathrm{.}\mathrm{.}\mathrm{.}\le a_n$, then there is a homogeneous ideal $J$ containing $x_1^{a_1},\mathrm{.}\mathrm{.}\mathrm{.},x_n^{a_n}$ with the same Hilbert function. In this paper we prove the Eisenbud-Green-Harris conjecture when $f_i$ splits into linear factors for all $i$.