On the regularity index of $s$ equimultiple fat points not on a linear   $(r-1)$-space, $s \le r+3$
  We prove the Trung's conjecture about Segre's upper bound for s equimultiple fat points not on a linear (r-1)-space, s\le r+3, by algebraic method used in [3]. This method also may used to research other cases of fat points. 
Phan Van Thien
math.AC