Fat Points in P^1 x P^1 and their Hilbert Functions

A bstract. W e study the H ilbert functions offat points in P 1 (cid:2) P 1 . If Z (cid:18) P 1 (cid:2) P 1 isan arbitrary fatpointschem e,then itcan be shown that forevery i and j the values of the H ilbert function H Z ( l;j ) and H Z ( i;l ) eventually becom e constant for l (cid:29) 0. W e show how to determ ine these eventualvalues by using only the m ultiplicitiesofthe points,and the relative positionsofthe points in P 1 (cid:2) P 1 . This enables us to com pute allbut a (cid:12)nite num ber values of H Z withoutusing the coordinatesofpoints.W e also characterize the ACM fatpointsschem esusing ourdescription ofthe eventualbehaviour.In fact,in the case that Z (cid:18) P 1 (cid:2) P 1 is ACM ,then the entire H ilbert function and its m inim alfree resolution depend solely on knowing the eventualvalues ofthe H ilbertfunction.