Let $V$ be a linear space, $V_1$, $V_2$ and $V_3$ be linear subspaces of $V$. Consider $\mathbb P(V/V_1)\times\mathbb P(V/V_2)\times\mathbb P(V/V_3)$ as a space parametrizing the triples of linear subspaces $(W_1, W_2,W_3)$, with $V_i\subset W_i\subset V$ and $\dim(W_i)-\dim(V_i)=1$ for $i=1,2,3$.

Now consider the closed subspace P consisting of the points $(W_1,W_2,W_3)\in \mathbb P(V/V_1)\times\mathbb P(V/V_2)\times\mathbb P(V/V_3)$, with $W_1\cap W_2\cap W_3\neq 0$.

Then the question is: now that P is a closed subscheme of $\mathbb P(V/V_1)\times\mathbb P(V/V_2)\times\mathbb P(V/V_3)$, how to compute trivariate Hilbert Polynomial of P without explicitly writing down the coordinates of P?

Let $V$ be a linear space, $V_1$, $V_2$ and $V_3$ be linear subspaces of $V$. Consider $\mathbb P(V/V_1)\times\mathbb P(V/V_2)\times\mathbb P(V/V_3)$ as a space parametrizing the triples of linear subspaces $(W_1, W_2,W_3)$, with $V_i\subset W_i\subset V$ and $\dim(W_i)-\dim(V_i)=1$ for $i=1,2,3$, and also $V_1\cap V_2\cap V_3=0$.

Now consider the closed subspace P consisting of the points $(W_1,W_2,W_3)\in \mathbb P(V/V_1)\times\mathbb P(V/V_2)\times\mathbb P(V/V_3)$, where $\bigcap_IW_i\supsetneqq \bigcap_IV_i$, with $I$ arbitrary subset of $\{1,2,3\}$.

Then the question is: now that P is a closed subscheme of $\mathbb P(V/V_1)\times\mathbb P(V/V_2)\times\mathbb P(V/V_3)$, how to compute trivariate Hilbert Polynomial of P without explicitly writing down the coordinates of P?