Towards Good Postulation of Fat Points, One Step at a Time

Abstract We give an overview on the landscape of polynomial interpolation theory. We will describe first the geometric approach, based on the base locus analysis of linear systems of hypersurfaces of $${\mathbb {P}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> with given degree and assigned multiplicity at a set of points. Secondly, we will consider the algebraic counterpart, with a discussion on the good postulation of fat point schemes of $${\mathbb {P}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> and their regularity index. In both cases we report on some complete, or partial, results and conjectures.