We will begin by reviewing the classical moduli theory of curves developed by Deligne–Mumford and Grothendieck–Knudsen. We will then discuss its higher dimensional analogues, i.e., the KSBA moduli theory. A central issue is that flatness behaves differently in higher dimensions: it is too weak for the underlying variety, but too strong for the divisorial part. Finally, using an arrangement of lines as an example, we will illustrate how the Minimal Model Program (MMP) can be used to find stable degenerations.

Several KSBA compactifications have been constructed explicitly, with their families (all stable pairs) described in detail. Examples include pairs of plane curves with small coefficients, principally polarized abelian varieties (PPAVs), hyperplane arrangements, K3 surfaces with recognizable divisors and so on. We will briefly review some of these constructions then focusing on hyperplane arrangements, where we will discuss the two different approaches developed by Hacking–Keel–Tevelev via visible contours and by Alexeev via geometric invariant theory (GIT). Finally, we will propose several problems for graduate students to explore. 

In 1966, Hartshorne proved that Hilbert schemes of projective space with a fixed Hilbert polynomial are connected. In contrast, Seong (2020) established disconnectedness of certain Hilbert schemes of ordinary Grassmannians and computed their nef cones in the hypersurface setting. Motivated by these results, this talk explores analogous questions for orthogonal Grassmannians. We will begin with a crash course in Schubert calculus, introducing Schubert varieties and the classical Pieri rule. We will then discuss Hilbert schemes parametrizing hypersurfaces in linear spaces contained in orthogonal Grassmannians, explaining how to describe their connected components and determine their nef cones.

Title : As a continuation of the previous talk, we will discuss the construction of the KSBA moduli spaces and the universal families for hyperplane arrangements. Kapranov’s version of Gelfand—MacPherson correspondence provides a natural candidate for the moduli space. Subsequently, Hacking—Keel—Tevelev and Alexeev constructed the corresponding families using matroid polytope tilings, via different approaches (visible contour and GIT respectively). We will also discuss wall-crossing problems in KSBA moduli theory, with particular emphasis on the case of hyperplane arrangements.

- Kyoung-Seog Lee (10:30)
- Rodolfo Aguilar Aguilar (2:00)