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Algebraic Geometry Seminar 23/24 Term 1

The algebraic geometry seminar in Term 1 2023/2024 will usually meet on Wednesdays at 3pm in B3.02, though we may sometimes change to allow speakers from other time zones.

See the talks from last term here.

Wednesday 4 October 2023, 3pm. Speaker: Charles Favre (École Polytechnique)

Title: b-divisors and dynamical applications

Abstract: b-divisors were introduced by Shokurov in the context of the minimal model program. We shall explain how to develop a positivity theory of these objects that have remarkable applications to algebraic dynamics.

Wednesday 11 October 2023, 3pm. Speaker: Calla Tschanz (University of Bath)

Title: Expanded degenerations for Hilbert schemes of points

Abstract: Let X –> C be a projective family of surfaces over a curve with smooth general fibres and simple normal crossing singularity in the special fibre X_0. We construct a good compactification of the moduli space of relative length n zero-dimensional subschemes on X\X_0 over C\{0}. In order to produce this compactification we study expansions of the special fibre X_0 together with various GIT stability conditions, generalising the work of Gulbrandsen-Halle-Hulek who use GIT to offer an alternative approach to the work of Li-Wu for Hilbert schemes of points on simple degenerations. We construct stacks which we prove to be equivalent to the underlying stack of some choices of logarithmic Hilbert schemes produced by Maulik-Ranganathan.

Wednesday 18 October 2023, 3pm. Speaker: Vaidehee Thatte (KCL)

Title: Understanding the Defect via Ramification Theory

Abstract: Classically, the degree of a finite Galois extension of complete discrete valuation fields equals the product of two invariants measuring the change in the valuation (ramification index) and the change in the residue field (inertia degree). More generally, there is a third factor - the ‘defect’. For example, we can have a degree p extension with trivial extensions of the value group and the residue field. The defect is not yet well understood and remains the main obstruction to several long-standing open problems in positive residue characteristic (e.g., resolution of singularities). The primary reason is, roughly speaking, that the classical strategy of "objects become nicer after finitely many blow-ups" fails when the defect is non-trivial. We are thrown into an infinite loop.

In this talk, I will discuss how techniques in arithmetic algebraic geometry, number theory, and valuation theory together can help us understand the defect and deal with the obstacles it creates. In particular, I will present a generalization of the classical invariants of ramification theory, a generalization of Gabber-Ramero's work on 'filtered union', and some recent work (joint with K. Kato) on upper ramification filtration in the general case. Any necessary background in these areas will be covered via examples and rough (practical) definitions.

Wednesday 25 October 2023, 3pm. Speaker: Farhad Babaee (Bristol)

Title: Complex tropical currents

Abstract: In this talk, I will recall basic ideas in tropical geometry and the theory of positive currents, and I will discuss why exploring the interactions of these two domains is natural and useful.

Wednesday 1 November 2023, 3pm. Speaker: Marvin Anas Hahn (Trinity College Dublin)

Title: Mustafin degenerations of syzygy bundles

Abstract: Mustafin varieties are degenerations of projective spaces, which are induced by point configurations in a Bruhat Tits building. In this talk, we use these degenerations to construct certain models of plane curves. Motivated by recent advances towards a p-adic Narasimhan—Sehsadri theorem, we then use these models to construct families of syzygy bundles which admit strongly semistable reduction. This talk is based on a joint work with Annette Werner.

Tuesday 7 November 2023, 3pm. Speaker: Alicia Dickenstein (Buenos Aires)

D1.07 (Note unusual date and location)

Title: Iterated and mixed discriminants

Abstract: Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schafli for the study of hyperdeterminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hypersurfaces was given by Ottaviani. In joint work with Sandra di Rocco and Ralph Morrison, we propose and study a generalization of this question for systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant. We define a related polynomial called the multivariate iterated discriminant. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre-Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.

Wednesday 8 November 2023, 3pm. Speaker: Shengyuan Huang (Birmingham)

Title: The orbifold Hochschild product for Fermat hypersurface

Abstract: For a smooth scheme X, the Hochschild cohomology of X is isomorphic to the cohomology of polyvector fields as algebras. This result is claimed by Kontsevich and then proved by Calaque and Van den Bergh. In this talk, I will present my recent progress with Andrei Caldararu and Kai Xu in generalising the result above to orbifolds.

In the projective spaces, one can consider the Fermat hypersurfaces with natural group actions. These are the main examples that we focus on in this talk. If we further assume that the hypersurface is Calabi-Yau, we prove the algebra isomorphism between its Hochschild cohomology and polyvector fields.

Wednesday 15 November 2023, 3pm. Speaker: Jeffrey Hicks (Edinburgh)

Title: Resolutions in toric varieties

Abstract: In this talk, I will discuss some recent work constructing resolutions of a toric variety X. Given a toric subvariety Y of codimension k, we will exhibit a resolution of the direct image of the structure sheaf of Y of length k; furthermore every term in this resolution will be a direct sum of line bundles. The resolution is constructed by computing the homology of the (real!) torus with coefficients in the category of coherent sheaves on X. I will mostly work through a specific example (resolving they skyscraper sheaf of a point on the projective plane) and describe how the proof that this provides a resolution arises from invariance of Morse homology under variation of Morse function. All work is joint with Andrew Hanlon and Oleg Lazarev.

Wednesday 22 November 2023, 3pm. Speaker: Sara Veneziale (LSGNT)

Title: Machine learning and the classification of Fano varieties

Abstract: In this talk, I will describe recent work in the application of AI to explore questions in algebraic geometry, specifically in the context of the classification of Fano varieties. We ask two questions. Does the regularized quantum period know the dimension of a toric Fano variety? Is there a condition on the GIT weights that determines whether a toric Fano has at worst terminal singularities? We approach these problems using a combination of machine learning techniques and rigorous mathematical proofs. I will show how answering these questions allows us to produce very interesting sketches of the landscape of weighted projective spaces and toric Fanos of Picard rank two. This is joint work with Tom Coates and Al Kasprzyk.

Wednesday 29 November 2023. No seminar!

Please note:

-- Patrick Kennedy-Hunt's talk has been postponed to Term 3.

-- Xenia de la Ossa's talk has been postponed to Jan 24th.

Wednesday 6 December 2023, 3pm. Speaker: Rudradip Biswas (Warwick)

Title: There is only one equivalence class of bounded t structures in the derived bounded category of coherent sheaves on a noetherian finite dimensional scheme.

Abstract: In the study of triangulated categories, bounded t-structures have always attracted a lot of attention. I will talk about one of my new papers, joint with Hongxing Chen, Kabeer Manali Rahul, Chris Parker, and Junhua Zheng, where we show that all bounded t structures on the derived bounded category of coherent sheaves on a noetherian finite dimensional scheme are equivalent to each other. Some major groundbreaking work in this area recently came from Amnon Neeman. Our paper generalizes his results because he could only prove this equivalence result when the scheme was separated and quasi excellent or had a dualising complex.