The regularity of points in multi-projective spaces

Let I=℘1m1∩⋯∩℘sms be the defining ideal of a scheme of fat points in Pn1×⋯×Pnk with support in generic position. When all the mi's 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.