On the Hartshorne–Hirschowitz theorem

The Hartshorne–Hirschowitz theorem says that a generic union of lines in Pn, (n≥3), has good postulation. proof Hartshorne and Hirschowitz the initial case P3 is handled by method specialization via smooth quadric surface with property having two rulings skew lines. We provide based on new degeneration disjoint plane H≅P2, which we call (2,s)-cone configuration, schematic s intersecting passing through single point P together trace an s-multiple supported at double 2H. In first part this paper, discuss our inductive approach. prove configuration P3, or more generally Pn. second use effective to show imposes independent conditions linear system |OP3(d)| surfaces given degree d. basic motivation behind approach it looks systematic gives some hope extensions analogous problem higher dimensional spaces, postulation for m-dimensional planes P2m+1.