Schedule
13:00-14:00 Milena Hering
14:00-14:30 Marina Godinho
14:30-15:00 Coffee Break
15:00-16:00 Ananyo Dan
16:00-16:15 Break
16:15-16:45 Marc Truter
16:45-17:00 Break
17:30-18:00 Alastair King
18:00-20:00 Dinner
Schedule
13:00-14:00 Milena Hering
14:00-14:30 Marina Godinho
14:30-15:00 Coffee Break
15:00-16:00 Ananyo Dan
16:00-16:15 Break
16:15-16:45 Marc Truter
16:45-17:00 Break
17:30-18:00 Alastair King
18:00-20:00 Dinner
TITLES AND ABSTRACTS
Milena Hering
Title: Equations of toric vector bundles
Abstract: The projectivisation of a very ample vector bundle E admits a natural embedding into projective space. We describe defining equations for this embedding in the case that the underlying variety and the vector bundle are toric. The main tool is to embedd P(E) into the projectivisation of a direct sum of line bundles F and to describe the equations in the Cox ring of P(F). This is joint work with Diane Maclagan and Greg Smith.
Marina Godinho
Title: New derived symmetries for varieties with (relative) tilting bundles
Abstract: A ring morphism p: A ⟶B satisfying certain mild assumptions induces a derived endomorphism of A and a derived endomorphism of B, which are closely related. In fact, the derived endomorphism of A is the twist around the restriction of scalars functor, and the derived endomorphism of B is the corresponding cotwist. I will discuss settings in which these endomorphisms are derived equivalences and use this technology to construct new derived autoequivalences of varieties with (relative) tilting bundles. We will construct some interesting examples of these new autoequivalence for quotient singularities.
Ananyo Dan
Title: Lefschetz (1,1)-theorem for singular varieties
Abstract: Lefschetz (1,1)-theorem states that every (1,1) class in a smooth projective variety is the first Chern class of a line bundle. Such a statement fails when the variety is singular. There have been various attempts at extending the Lefschetz (1,1) to singular varieties. The most universal statement so far is due to Arapura. However, it follows from the work of Totaro that the map studied by Arapura does not look at all the possible (1,1) classes, in the singular case. Totaro suggests looking at the Bloch-Gillet-Soule cycle class map from the operational Chow group to the space of Hodge classes. In a joint work with I. Kaur, we study this map and give a criterion under which this map is surjective, thereby giving a possible Lefschetz (1,1) theorem for singular varieties. In the talk, I plan to present these results and give various examples where the surjectivity holds.
Marc Truter
Title: Fano 4-fold hypersurfaces
Abstract: In 2016, Birkar proved that there are finitely many Fano varieties with terminal singularities in each dimension. This finiteness result motivates the construction of 'periodic tables' of such varieties, analogous to those in dimensions 1 and 2, with great progress made in dimension 3, but where dimension 4 remains largely uncharted territory.
Motivated by this, we study Fano 4-fold hypersurfaces with terminal singularities. A key departure from the 3-dimensional theory is that quasismoothness is no longer a reasonable generality assumption, and the resulting singularities are significantly wilder than in dimension 3. To navigate this nonquasismooth setting, we develop a finite-step criterion for terminality, refining the classical infinite-step criterion of Mori. The power of the toolbox we develop is demonstrated by producing over 100,000 new examples, including previously intractable high degree cases, contributing to the emerging periodic table in dimension 4.
Alastair King
Title: Mysterious Duality and helical line bundles on del Pezzo surfaces.
Abstract: Vafa’s Mysterious Duality is a correspondence between (classes of) certain rational curves on del Pezzo surfaces and (charges of) fields in Type II supergravity. I will explain how the line bundles associated to the curves have a natural characterisation and how both sides of the correspondence are related to gradings of the Lie algebra E8. The talk is based on arXiv:2507.10169.