Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions

Abstract We study the Hilbert functions of fat points in ℙ 1 × ℙ 1 . If Z ⊆ ℙ 1 × ℙ 1 is an arbitrary fat point scheme, then it can be shown that for every i and j the values of the Hilbert function H Z ( l , j ) and H Z ( i , l ) eventually become constant for l ≫ 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ℙ 1 × ℙ 1 . This enables us to compute all but a finite number values of H Z without using the coordinates of points. We also characterize the ACM fat point schemes using our description of the eventual behaviour. In fact, in the case that Z ⊆ ℙ 1 × ℙ 1 is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.