Interpolation in Weighted Projective Spaces

Ponente(s): Shahriyar Roshan Zamir Hatami

Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree d, singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992--1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this talk, we primarily use commutative algebra to prove analogous statements in the weighted projective space, a natural generalization of the projective space. We will also introduce an inductive procedure, originally due to A. Terracini from 1915, to demonstrate an example of a weighted projective plane, where the analogue of the Alexander-Hirschowitz theorem holds without any exceptions. Furthermore, we will give interpolation bounds for an infinite family of weighted projective planes. There are no prerequisites for this talk besides some elementary knowledge of commutative algebra.