Hilbert function of points in $\mathrm{P}^2$

Question

Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If $$ \mathrm{gin_{rlex}}(I)=(x_1^k,x_1^{k-1}x_2^{\lambda_{k-1}},...,x_1x_2^{\lambda_1},x_2^{\lambda_0} ), $$ then how we can show , $h_{S/I}(m)= \sum_{i=0}^{k-1}\mathrm{min}(m-i,\lambda_i)$?

Answer 1

One can show that for every $m$ with $I_m\neq 0$, after a generic change of coordinates, $$I_m = \langle f_{m1},\dots, f_{mj_m}\rangle \oplus x_3I_{m-1}, $$ s.th. the initial terms of $f_{m1},\dots, f_{mj_m}$ are equal to $\{x_1^m, x_1^{m-1}x_2,\dots,x_1^{m+1-j_m}x_2^{j_m-1}\}$. (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 $h_{S/I}(m) = h_{S/I}(m-1) + m+1-j_m$, and $$m+1-j_m = \mathrm{min}\{i:i+\lambda_i \leq m\}.$$ Using these facts and induction on $m$, we can prove the following formula: $$h_{S/I}(m)=\sum_{i=0}^{\mathrm{min}(k,m)}\mathrm{min}(m-i+1,\lambda_i). $$