Bologna-Waseda Lectures
on Algebraic Geometry

• Date: Sep. 7 – Sep. 9, 2026
• Place: Room 51-18-12, Nishi-Waseda Campus, Waseda University.

• Short Courses by
  • Enrico Fatighenti “Birational and biregular geometry of Fano varieties in low dimensions”
  • Giovanni Mongardi “Rational curves on IHS manifolds and their applications to Chow rings”
  • Andrea Petracci “Deformations of toric varieties and applications”

Schedule

	10:00 – 11:40	13:30-15:10
Sep. 7 (Mon.)	Fatighenti (1)	Mongardi (1)
Sep. 8 (Tue.)	Petracci (1)	Fatighenti (2)
Sep. 9 (Wed.)	Mongardi (2)	Petracci (2)

Abstract

[Fatighenti] Fano varieties are among the fundamental building blocks of higher-dimensional algebraic geometry, but their classification rapidly becomes more intricate as the dimension and the Picard rank increase. In these lectures, I will present an informal introduction to the classification of Fano varieties, emphasizing the distinction between birational and biregular models.Starting from elementary examples, we will discuss the biregular classification of del Pezzo surfaces and then move to Fano threefolds, including Mukai’s models and the interpretation of higher-Picard-rank threefolds as zero loci of vector bundles on suitable ambient varieties. This will provide the background for the four-dimensional case, where no complete classification is currently available.
We will then discuss several strategies for studying Fano fourfolds and introduce the FanoBase project, its current scope, and some of the birational geometry emerging from it. Time permitting, I will conclude with applications to varieties of K3 type arising from FanoBase, together with related questions concerning degenerations and even nodal surfaces of K3 type.

[Mongardi] In these lectures, we give a survey of the theory of rational curves in IHS manifolds: we will describe their deformation theory and focus on rational curves whose deformations cover a divisor inside the IHS manifold.
We will give a proof of the existence of these ruled divisors in the greatest generality, and we will give some techniques to prove that points lying on a ruled divisor are dense (in some specific deformation classes).

[Petracci] Toric varieties lie at the intersection of algebraic geometry and of combinatorics. In these lectures I will review the basics of toric varieties and of toric singularities and I will survey some facts about their deformation theory, with applications to moduli of Fano varieties.

• Organizer: Yasunari Nagai (nagai.y [at] waseda.jp)