The regularity of points in multi-projective spaces

Let I= 1m1∩ ∩ sms be the defining ideal of a scheme of fat points in with support in generic position. When all the miʹs are 1, we explicitly calculate the Castelnuovo–Mumford regularity of I. In general, if at least one mi 2, we give an upper bound for the regularity of I, which extends a result of Catalisano, Trung and Valla.