Algebraic Geometry Seminar
The algebraic geometry seminar in Term 3 2024/2025 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.
Thursday 1 May 2025, 3pm. B3.03 Speaker: Ruadhaí Dervan (Glasgow/Warwick) (Note unusual time and location!)
Title: K-stability and moduli of higher-dimensional varieties
Abstract: The construction of the moduli space of stable curves is a cornerstone of algebraic geometry, and it is natural to ask whether this construction can be generalised to higher-dimensional projective varieties. While many new difficulties arise, the theory of K-stability is conjecturally the right tool to construct such moduli spaces. The construction of moduli of K-stable Fano varieties has been, over the last decade, fully understood, but we know very little for more general varieties. I will discuss some recent progress in this direction using a variant of K-stability, partially joint work with Rémi Reboulet.
Wednesday 7 May 2025, 3pm. B3.02 Speaker: Christian Böhning (Warwick)
Title: Fourier-Mukai partners of non-syzygetic cubic fourfolds and Gale duality
Abstract: In this project we study so-called non-syzygetic cubic fourfolds, i.e., smooth cubic fourfolds containing two cubic surface scrolls in distinct hyperplanes with intersection number between the two scrolls equal to 1. We prove that a very general non-syzygetic cubic fourfold has precisely one nontrivial Fourier-Mukai partner that is also non-syzygetic. We characterise non-syzygetic cubic fourfolds algebraically as those having a special type of equation that is almost linear determinantal, and show that the equation of the Fourier-Mukai partner can be obtained by applying Gale duality. We establish that Gale dual cubics are birational, Fourier-Mukai partners and have birational Fano varieties of lines under suitable genericity assumptions. We show that the birationality of the Fano varieties of lines continues to hold in the context of equivariant birational geometry, but birationality of the cubics may not. We exhibit examples of Gale dual cubics with faithful actions of the alternating group on four letters that could provide counterexamples to equivariant versions of a conjecture by Brooke-Frei-Marquand predicting birationality of the cubics if the Fano varieties of lines are birational, and also possibly a related conjecture by Huybrechts predicting birationality of Fourier-Mukai partners.
Joint work with Lisa Marquand and Hans-Christian von Bothmer.
Thursday 15 May 2025, 1:30-5:30 pm: LMS Tropical Mathematics Network Meeting (All talks in MB0.07)
3 talks with coffee breaks in between:
1:30-2:30 pm: Hannah Markwig (Tübingen)
Title: Tropical curve counting
3-4pm: Stefano Mereta (KTH)
Title: The space of valuated preorders and the congruence spectrum of $S[x_1, \dots , x_n]$ as the spectrum of a ring
4:30-5:30 pm: Alex Esterov (LIMS)
Title: Schön complete intersections
Abstracts:
Hannah Markwig: In enumerative geometry, we fix geometric objects and conditions and count how many objects satisfy the conditions. For example, there are 2 plane conics passing through 4 points and tangent to a given line. Tropical geometry can be viewed as a degenerate version of algebraic geometry and has proved to be a successful tool for enumerative problems. We review tropical curve counting problems. In particular, we show how tropical methods can be applied for quadratically enriched counts, which can be viewed as generalizations that allow results over any ground field.
Stefano Mereta: In the late '60s, Hochster introduced the notion of spectral space and proved that every spectral space is the spectrum of a commutative ring. Unfortunately the proof of this fact is not constructive. More recently, Jun, Ray and Tolliver have proven constructively that every spectral space is the $k$-spectrum of an idempotent semiring, and that the congruence spectrum of an idempotent semiring is a spectral space.
In this talk we will introduce the notion of a valuated preorder on the monomials of $K[x_1, \dots , x_n]$ for a valued field $K$ and realise it, relying partially on work by Jóo and Mincheva, as the $k$-spectrum of a semiring, as the congruence spectrum of a polynomial semiring and, most importantly, as the spectrum of a commutative ring.
This commutative ring is constructed as the "unit ball" of respect a generalised Bézout valuation.
Alex Esterov:
There is a number of "aesthetically similar" topics in combinatorial algebraic geometry, such as toric complete intersections, hyperplane arrangements, simplest singularity strata of general polynomial maps, some discriminant and incidence varieties in enumerative geometry and polynomial optimization, polynomial ODEs such as reaction networks, generalized Calabi--Yau complete intersections.
I will talk about a convenient umbrella generality for all of them, which still admits a version of the classical theory of Newton polytopes (but with so-called tropical complete intersections instead of polytopes).
Wednesday 28 May 2025, 3pm. Speaker: Timothy Logvinenko (Cardiff)
Title: Perverse schobers and the McKay correspondence
Abstract:
I will report on the ongoing project to construct a perverse schober, a poor man’s perverse sheaf of triangulated categories, in the context of the classical two-dimensional McKay correspondence for G \subset SL_2(C). The braid group of the corresponding ADE type acts on the derived category D(Y) of the minimal resolution Y of C^2/G by spherical twists in the exceptional curves. Since this braid group is the fundamental group of the open stratum of \mathfrak{h}/W, the quotient of the ADE Cartan algebra by the Weil group action, its action on D(Y) can be thought of as a local system of triangulated categories with the fiber D(Y) on this open stratum. A perverse schober extends this structure to the higher codimension stratas.
We actually construct a W-equivariant schober on \mathfrak{h} by using an instance of the McKay correspondence – the root hyperplane arrangement in \mathfrak{h} coincides with the wall-and-chamber structure in the stability space \Theta for the GIT construction of Y as the moduli space of \theta-stable G-constellations. We can thus make use of the techniques of Halpern-Leistner-Sam and Spenko-Van-den-Bergh for the quasi-symmetric reductive group action on a vector space, though in our case the group acts on a very singular subvariety of a quasi-symmetric vector space.
Our work is motivated by wanting to eventually tackle dim=3 case, where h/W picture no longer exists and the GIT action for G-constellation moduli is no longer quasi-symmetric. However, it might still be possible to construct a schober on the GIT stability space, neatly packaging up all the Craw-Ishii GIT wall-crossing equivalences and more. This is a joint work with Arman Sarikyan (LIMS).
Wednesday 11 June 2025, 3pm. Speaker: Lisa Marquand (NYU)
Title: Symplectic birational transformations of OG10 Hyperkähler manifolds
Abstract: New examples of compact hyperkähler manifolds are notoriously hard to construct. One approach is to consider a hyperkähler manifold of known type that admits a finite group of symplectic automorphisms, i.e. automorphisms preserving the holomorphic symplectic form. In this case, both the fixed locus and quotient will be symplectic varieties - one hopes to obtain something new (even if singular). The difficulty with this program is twofold: one needs to find a promising group action, and also a geometric example that one can study. In order to do so, often one needs to broaden the search to finite groups of symplectic birational transformations. In this talk I will describe how to first classify possible symplectic group actions for O'Grady's 10-dimensional example, based on joint work with Stevell Muller. I will give examples where it is possible to study the fixed locus for such an action, utilising the construction of Laza, Saccà and Voisin, starting from a cubic fourfold with automorphisms. If time permits, I will report on ongoing work to identify the fixed locus in such an example.