Algebraic Geometry Seminar
The algebraic geometry seminar in Term 1 2025/2026 will usually meet on Wednesdays at 3 pm, though we may sometimes change to allow speakers from other time zones.
Most talks will be either in B3.02 or MB0.07 -- see specific information below.
A link to talks from the previous term is here.
Wednesday 8 Oct 2025, 3pm. Speaker: Alessio Cela (Cambridge)
Title: Tangent Bundles and fixed domain curve counts
Abstract: In this talk, I will explain how the non-vanishing of fixed-domain curve counts implies the stability of the tangent bundle TX with respect to the curve class in question. I will then apply this result to establish the vanishing of most Tevelev degrees for Hirzebruch surfaces and compute the remaining cases by relating them to the corresponding invariants of the projective line. Time permitting, I will also explain how to compute the splitting type of the restriction of the tangent bundle of blow-ups of projective space along a general rational curve, via fixed-domain curve counts. This is joint work with Carl Lian.
Wednesday 22 Oct 2025, 3pm. Speaker: Hao Zhang (Glasgow)
Title: Local forms for the double An quiver and Gopakumar–Vafa invariants
Abstract: This talk concerns the birational geometry of cAn singularities and their curve-counting invariants through noncommutative methods. I introduce generalised GV invariants for crepant partial resolutions and verify Toda’s formula in this setting. Turning to crepant resolutions, I give intrinsic definitions of Type A potentials on the doubled An quiver (with a loop at each vertex). After coordinate changes, these admit a monomialized form and classify all crepant resolutions of cAn, confirming the Brown–Wemyss Realisation Conjecture; an explicit example shows that the Donovan–Wemyss Conjecture fails in the non-isolated case. Building on the correspondence between crepant resolutions and Type A potentials, I give numerical constraints on possible GV tuples and an explicit classification for cA2. Time permitting, I will also mention the n ≤ 3 classification of Type-A loop-free potentials up to isomorphism and derived equivalence. This is based on arXiv:2412.10042 and arXiv:2504.03139.
Wednesday 29 Oct 2025, 3pm. Speaker: David Eisenbud (Berkeley)
Title: A survey of (mostly infinite) free resolutions
Abstract: Finite free resolutions, such as those over polynomial rings, have been studied since the late 19th century, and have well-established applications in algebraic geometry. But most modules over most rings have only infinite resolutions. Among other applications, these arise in the study of group cohomology, and, apparently for this reason, a certain class of infinite resolutions were analyzed by John Tate in a groundbreaking paper of 1957. This and a result of Serre's led to the "Serre-Kaplansky" problem, which dominated the field until 1982.
Recently I and others have been investigating these resolutions from a new point of view. I'll explain the old and new conjectures, and give an historical overview of the field.
Wednesday 5 Nov 2025, 3pm. Speaker: Thamarai Venkatachalam (UCL)
Title: Classification of Q-Fano 3-folds.
Abstract: Fano 3-folds with terminal singularities are fundamental building blocks in birational geometry, but their classification remains an important open problem. In this talk, I will discuss my work on the classification of Fano 3-fold hypersurfaces with terminal singularities embedded in rank 2 toric varieties, viewed as GIT quotients. By exploiting this toric structure and the Minimal Model Program, I provide a complete list of families in my setup and relate them to existing results on Fano varieties with terminal singularities.
Wednesday 12 Nov 2025, 3pm. Speaker: Lars Martin Sektnan (University of Gothenburg)
Title: Stability of projective bundles
Abstract: K-stability is a notion stability for varieties that has become important to form moduli of varieties and is closely related to the existence of canonical Kähler metrics on the variety through the Yau-Tian-Donaldson conjecture. We consider K-stability on projective bundles, with respect to so-called adiabatic polarisations. Older works of Hong and Ross-Thomas show that K-stability for these polarisations is closely related to slope stability of the underlying vector bundle. Our main result is that K-stability of the projective bundle is equivalent to a notion of stability for the vector bundle that we call asymptotic slope stability. This stability notion has similar properties, but is not equivalent, to Gieseker stability. We use our result it to find bundles that are not slope stable, but whose corresponding projective bundle is K-stable. This is joint work with Annamaria Ortu
Wednesday 19 Nov 2025, 3pm. Speaker: Stefania Vassiladis
Title: Explicit bounds on foliated surfaces and the Poincaré problem
Abstract: We give a solution to the Poincaré Problem, in the formulation of Cerveau and Lins Neto. We obtain a bound on the degree of general leaves of foliations of general type, which is linear in the genus of the leaf. To achieve this, we will introduce the notion of foliation. I will give some examples that show some pathologies that occurs when dealing with foliations, and how to deal with them using adjoint foliated structures.
Wednesday 26 Nov 2025, 3pm. Speaker: Ben Davison (Edinburgh)
Title: Tutte polynomials of graphs and symplectic duality
Abstract: The Tutte polynomial of a graph is a two-variable polynomial, which is the universal polynomial satisfying deletion contraction recursion. In this talk, I will explain how this polynomial arises from considering the cohomology of hypertoric varieties (which I'll introduce) along with the two (not one!) filtrations by cohomological degree, coming from symplectic duality (which I will also introduce).
Thursday 27 Nov 2025, 2 - 6pm.
COW seminar Schedule
2 pm Speaker: Ben Davison (Edinburgh)
Title: The topology of the stack of semistable coherent sheaves on a K3 surface
Abstract: Let nu be a Mukai vector for a K3 surface S. When nu is primitive, the moduli space of semistable coherent sheaves (with respect to a generic polarisation) is deformation equivalent to a Hilbert scheme of points on the same K3 surface. This moduli space is smooth, and its Poincaré polynomial is known by Göttsche's formula. The topology of the stack of semistable sheaves is then determined by the topology of this moduli space. By a theorem of Markman, the cohomology is generated by tautological classes.
3 pm – Discussion break
4 pm Speaker: Daniel Huybrechts (Bonn)
Title: The period-index problem for hyperkähler varieties
Abstract: Brauer-Severi varieties (so smooth fibrations with projective spaces as fibres) are not Zariski locally trivial. The failure is measured by the associated Brauer class to which two numerical invariants are attached: period and index. The precise relation between the two is unknown but the index is conjectured to be universally bounded by some power of the period. The talk will start with a gentle introduction into the general theory with a survey of things that are known. In the second half I will study the problem for hyperkähler varieties in which case a better bound is expected and in fact can be proved in interesting cases.
5 pm Speaker: Jarek Buczyński (IMPAN, Warsaw)
Title: Three stories of Riemannian and complex projective manifolds
Abstract: Complex projective manifolds and Riemannian manifolds invite you all to participate in their three epic stories.
In the first tale, the main character is going to be a complex projective manifold, and as in every story, there will be some action going on. More specifically, the group of invertible complex numbers, or even better, several copies of those, act on the manifold. The spirit of late Andrzej Białynicki-Birula until this day helps us to comprehend what is going on.
The second story is a tale of holonomies, it begins with "a long time ago,..." and concludes with "... and the last missing piece of this mystery is undiscovered till this day". The protagonist of this part is a quaternion-Kähler manifold, while the legacy of Marcel Berger is in the background all the time.
In the third part, we meet legendary distributions, which are subbundles in the tangent bundle of one of our main characters. Among others, distributions can be foliations, or contact distributions, which like yin and yang live on the opposite sides of the world, yet they strongly interact with one another. Ferdinand Georg Frobenius is supervising this third part.
Finally, in the epilogue, all the threads and characters so far connect in an exquisite theorem on the classification of low-dimensional complex projective contact manifolds. In any dimension, the analogous classification is conjectured by Claude LeBrun and Simon Salamon, while in low dimensions it is proved by Jarosław Wiśniewski, Andrzej Weber, in a joint work with the narrator.
Wednesday 3 Dec 2025, 3pm. Speaker: Jarek Buczyński (IMPAN, Warsaw)
Title: Secant varieties of toric varieties, multigraded Hilbert scheme, and apolarity
Abstract: Given a smooth projective toric variety $X\subset\mathbb{P}^N$ its $r$-th secant variety is the closure of the union of linear spaces spanned by $r$ distinct points of $X$. We will present an elementary method to determine if a point of the projective space is in a given secant variety or not. The method is analogous to a classical theory called apolarity lemma. It can be used to describe all point of a secant variety uniformly, including those very special ones that resisted a systematic approach. We also define a secant variety version of the variety of sums of powers, which roughly encode all the possible ways a point can be presented as an element of the secant variety. We analyse its usefulness to study points with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some interesting tensors.
The talk is primarily based on https://arxiv.org/abs/1910.01944, joint with Weronika Buczyńska, Duke Mathematical Journal.
Wednesday 10 Dec 2025, 3pm. Speaker: Sema Güntürkün (Essex)
Title:
Abstract: