Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If $$ \mathrm{gin_{rlex}}(I)=(x_1^k,x_1^{k1}x_2^{\lambda_{k1}},...,x_1x_2^{\lambda_1},x_2^{\lambda_0} ), $$ then how we can show , $h_{S/I}(m)= \sum_{i=0}^{k1}\mathrm{min}(mi,\lambda_i)$?

2$\begingroup$ Can you explain more about the notations above? $\endgroup$– MostafaAug 2 '14 at 11:51

1$\begingroup$ Of course, $\mathrm{P}^2$ is projective plane in variables $x_1,x_2,x_3$. $\mathrm{gin}_{rlex}(I)$, denotes the generic initial ideal of ideal $I$, with respect to reverse lexicographic order, and $h_{S/I}(m)= \mathrm{dim}(S/I)_m$, is the Hilbert function of $S/I$ in degree m. $\endgroup$– A.B.Aug 2 '14 at 13:30
One can show that for every $m$ with $I_m\neq 0$, after a generic change of coordinates, $$I_m = \langle f_{m1},\dots, f_{mj_m}\rangle \oplus x_3I_{m1}, $$ s.th. the initial terms of $f_{m1},\dots, f_{mj_m}$ are equal to $\{x_1^m, x_1^{m1}x_2,\dots,x_1^{m+1j_m}x_2^{j_m1}\}$. (for this we can use only that no point is on the hyperplane $x_3=0$ and a generic coordinate change of the form $x'_1=x_1,x'_2=ax_1+x_2,x'_3=x_3$.)
Then we have $h_{S/I}(m) = h_{S/I}(m1) + m+1j_m$, and $$m+1j_m = \mathrm{min}\{i:i+\lambda_i \leq m\}.$$ Using these facts and induction on $m$, we can prove the following formula: $$h_{S/I}(m)=\sum_{i=0}^{\mathrm{min}(k,m)}\mathrm{min}(mi+1,\lambda_i). $$