Description:

In this series of lectures, we will study polynomial interpolation problems consisting of determining the dimension of linear systems of hypersurfaces of fixed degree and prescribed multiplicity at finite sets of points in a projective space of arbitrary dimension. In particular, we will be interested in computing the Castelnuovo-Mumford regularity of the ideal sheaf of fat point schemes, that is the smallest degree in which the corresponding linear system is not special, namely it has the expected dimension. The geometric approach to this problem, which is still open in its full generality, dates back to the Italian school of algebraic geometry; particularly relevant is work of Guido Castelnuovo, some of which we will aim at reading.

Moreover, we will consider a number of related topics as detailed in what follows.

L1: Plane singular curves. In the first lecture, we will focus on the planar case. We will state two well celebrated conjectures. Firstly, Segre’s conjecture on the speciality of linear systems of singular plane curves (later reformulated by Gimigliano and by Harbourne and Hirschowitz). Secondly, Nagata’s conjecture on the emptiness of these linear systems. We will study known results in this direction, some of which due to Castelnuovo.

L2: Nodal hypersurfaces. The second lecture will be dedicated to the higher dimensional case, with particular attention to the case of hypersurfaces with nodal singularities. We will state the Alexander-Hirschowitz theorem that solves the latter case and it is today the only complete classification result. Moreover, we will make a connection to the study of classical objects such as secant varieties of Veronese embeddings.

L3: Base loci. For arbitrary multiplicities, we will analyse the base loci of the linear systems and introduce the concept of special effect varieties: these are varieties that, whenever contained in the base with large multiplicity, cause the linear system to be special. A complete description is available in the case of up to n+3 points in n-dimensional space, or for an arbitrary number of points with bounded multiplicity, that is obtained by means of a cohomological analysis of the associated line bundles.

L4: Birational properties of blow-ups. We will study the birational geometry of projective spaces blown-up at a finite set of points and the relevant cones of divisors (effective, ample, nef). We will see how elementary (-1)-curves and (-1)-divisors (in the sense of Mukai) are special effect for the linear systems associated with effective divisors. Finally, we will see which of these blow-ups have finitely generated Cox ring (aka Mori dream spaces), making a link to Hilbert’s 14 th problem.