The course aims to teach the classical techniques for parameterising certain locally closed subschemes of the Hilbert and Quot schemes of points, namely Hilbert–Samuel strata. There will be a particular focus on using the software Macaulay2 in algebraic geometry. Five topics will be covered in five lectures during the course. An additional lecture will be devoted to explicit computations using the Macaulay2 software. Lecture notes written in collaboration with Dott. Gessica Alecci are available via the following link NotesCiAG.pdf.
Organisation of the course
(1) Classical projective and birational geometry
In the first lecture I will recall the definitions of classical maps in algebraic geometry such as blow-ups, Veronese/Segre/Plucker embeddings and the Cremona transformation. These classical constructions are essential for describing Hilbert–Samuel strata.
(2) Monomial ideals and partitions
In the second lecture, I will introduce some useful tools for working with ideals in the polynomial ring. These include the notions of socle and Macaulay duality. Along the way, we will see many examples concerning monomial ideals.
(3) Commutative algebra and families of ideals
In this lecture I will introduce the concept of a (flat) family of subschemes and its relation to that of a Hilbert polynomial. I will then present explicit constructions of families, which can be understood as morphisms to the Hilbert scheme.
(4) Hilbert schemes
This lecture will focus on the Hilbert scheme of points on a smooth variety. More specifically, we will study its tangent space and the corresponding Bialynicki-Birula decomposition. This tool provides a way to certify the presence of certain irreducible components of the Hilbert scheme, such as reduced elementary ones.
(5) The Grothendieck ring of varieties
In this lecture I will introduce the concept of a motive of an algebraic variety. Although motives are insensitive to many geometric properties of the corresponding object, I will explain how information can be extracted from them through explicit computations.
(6) Computations via Macaulay2
This lecture will serve as a proof of concept of the previous ones. I will show how to solve some of the exercises presented during the course. Finally, I will explain how to perform computations using the Macaulay2 package HilbertAndQuotSchemesOfPoints.m2 by Paolo Lella.