On the base locus of linear systems of general double points

Fix integers n ≥ 1, d ≥ 4 and  x>0 such that (n+1)(x-1) +Bin(n+2, 2) ≤ Bin(n+d, n). Take a general S ⊂ Pn such that #S=x and let B denote the scheme-theoretic base locus of |I2s(d)|, where 2S is the union of the double points with S as their reduction. Then 2S is the union of the connected components of B containing at least one point of S. We prove this theorem proving that a general union of x-1 double points and one triple point has no higher cohomology in degree d.