Products of homogeneous forms

Here we give a geometric proof of the following result. Let K be an algebraically closed field. Fix an integer sgreater-or-equal, slanted1 and positive integers ni and di, 1less-than-or-equals, slantiless-than-or-equals, slants. Set mi=min{ni,di+1}. For 1less-than-or-equals, slantiless-than-or-equals, slants and 1less-than-or-equals, slantjless-than-or-equals, slantni, take general homogeneous forms Fijset membership, variantK[x,y] with deg(Fij)=di. Let Iisubset ofK[x,y] be the homogeneous ideal generated by the forms Fij, for 1less-than-or-equals, slantjless-than-or-equals, slantni. Let dcolon, equals∑i=1s di and denote by (I1cdots, three dots, centeredIs)d be the degree d part of the homogeneous ideal I1cdots, three dots, centeredIs. Thenimage