1/23/2023 – Wern Yeen Yeong: How complicated (or irrational) is this variety?
A (smooth complex projective) variety is rational if it is birational to projective space. If a variety is not rational, there are many well-studied notions to describe how close it is to being rational, e.g. stably rational, unirational, rationally connected. Interesting examples of threefolds due to Clemens–Griffiths, Iskovskikh–Manin, Artin–Mumford show that these notions are not all secretly equivalent. Besides these notions, there is a growing interest in bounding some quantitative measures of irrationality, e.g. covering gonality and degree of irrationality. These measures have been studied/known for very general hypersurfaces and complete intersections in projective space, K3 surfaces, abelian varieties, etc. In this talk, we start by defining and comparing these notions and measures via examples. Then, we discuss a few common techniques used in these proofs, some of which like the Cayley–Bacharach property may be of independent interest to the audience. We finish by discussing (or listing) some interesting open problems in this area.
2/6/2023 – Matt Weaver: Jouanolou duality and the equations of Rees algebras
One of the classical problems within algebraic geometry is to determine the implicit equations defining the closed image and the graph of a rational map between projective spaces. Algebraically, this corresponds to determining the defining equations of the Rees algebra of the ideal whose generators define such a map. In this talk, we recount some recent developments in this area and how a classical technique can be combined with modern methods to describe these equations.
2/20/2023 – Emanuela Marangone: Inverse Systems and some applications
Given a homogeneous ideal I in the polynomial ring R=k[x0, x1,…, xn], the Macaulay’s Inverse System of I, denoted by I-1, is the submodule of S=k[y0, y1,…, yn] consisting of all the element of S annihilate by I, where we consider the apolarity action of R on S. In the first part of the talk we will discuss some properties of the apolarity action and of Inverse Systems.
In the second part, we will focus on ideals that define zero-dimensional schemes of fat points on Pn. The inverse System gives us a way to compute their Hilbert function as the dimension of an ideal of powers of linear forms.
Finally, in the last part, we will discuss how this allows us to connect the questions about unexpected cones with the Weak Lefschetz property. Precisely, given a finite set of points P1,…, PS in Pn, the points P such that there is an unexpected cone in degree d correspond to the linear forms LP that the multiplication map xLP on R/(LP1d,… ,LPsd) from degree d-1 to degree d does not have maximum rank.
3/06/2023 – Nikolai Konovalov: A few words about Jacobians
I will discuss polarized Hodge structures on cohomology groups of smooth projective varieties, intermediate Jacobians, and their applications to
some problems of rationality.