PhD projects
General interests: Foliation singularities, higher dimensional algebraic geometry, K3 surfaces, algebraic cycles and deformation theory, moduli spaces of curves and Gromov-Witten theory.
Dr Thomas Eckl
1. Jets to singularities of complex foliations
The PhD project should investigate the existing notions of good plane foliation singularities by means of jets and arcs tangent to the foliation. On examples this can be done by using computer algebra programs relying on Groebner basis techniques, like MACAULAY or in simpler cases, MAPLE. The results of these computations should help to predict statements for the general case. The model of such general results is Mustataís recent characterization of ordinary singularities of special types by jets and arcs. Finally one could try to transfer these statements to higher-dimensional cases, for example on foliations of the 3-dimensional complex space by 2-dimensional leaves. Their singularities were recently classified and resolved by Cano.
2. Classification of higher dimensional algebraic varieties
3. Nagata's conjecture on plane algebraic curves
Dr Vladimir Guletskii
Direction 1. Regular models, good and bad reduction of algebraic surfaces over number fields. Arithmetic threefolds arising from relative curves over bi-Dedekind schemes. Bertini-type theorems in arithmetic setup (after Jannsen-Saito and Poonen).
Direction 2. Schemes of rational curves on symmetric powers of surfaces, in zero and positive characteristics. The basic examples would be unirational supersingular K3s in positive characteristics (Liedtke, 2015), and surfaces of general type with p_g=0 in char 0, such as the Craighero-Gattazzo or Godeaux surfaces.
Direction 3. Spaces of 0-cycles, constructed as group completions (in the category of sheaves on the Nisnevich site) of infinite symmetric powers of schemes over a base. The objective here is to understand what would be the correct notion of a sufficiently deformable rational curve on the space of 0-cycles.
Direction 4. The study of Gysin kernels, i.e. the kernels of push-forward homomorphisms from the Chow group of 0-cycles on a smooth hyperplane section of a variety X, embedded into a projective space, to the Chow group of 0-cycles on X, in terms of rational curves on the spaces of 0-cycles of the variety X (see Direction 3). The etale monodromy argument.
Direction 5. Transcendental motives of smooth projective surfaces, their rational decomposability and integral indecomposability, and the relevance of the latter to birational geometry of 4-fold hypersurfaces in P^5. The degeneration argument in terms of transcendental motives. The arithmetic aspects of the story.
More details are given on the web-page: http://pcwww.liv.ac.uk/~guletski/
Naoki Koseki
1. Enumerative geometry
Gopakumar-Vafa invariants (also known as BPS invariants) provide a new curve counting theory. At the moment, there are only very little computational tools. In this project, we first try to calculate those invariants in new examples. We will then try to find a new structural result in the Gopakumar-Vafa theory.
2. Bridgeland stability conditions
Bridgeland stability conditions appear in many fields, including algebraic geometry, symplectic geometry, and representation theory. The set of all Bridgeland stability conditions form a complex manifold, and it is interesting to understand its topological properties (for example, one may ask when this space is non-empty). There are many possible approaches depending on one’s interest and background.
Dr Nicola Pagani
1. Stability spaces for line bundles on Spin/Prym curves
In a series of recent works, Fava-Pagani-Viviani introduced a new class of compactified Jacobians, called V-compactified Jacobians, and they proved that over the moduli space of stable n-pointed curves of genus g these are all existing (modular) compactifications of the Jacobian. Furthermore, they gave in [2603.05455] A complete classification of modular compactifications of the universal Jacobian an explicit description of the stability space for such Jacobian as a poset whose maximal element coincide with the compactified Jacobians that are also fine moduli spaces. This project is about extending these results to the cases of Spin and Prym universal moduli spaces, which have recently attracted a lot of attention for their connection to mirror symmetry and to Hurwitz theory respectively.
2. Wall-crossing of cup product on compactified Jacobians of a single curve
By work of Migliorini-Shende-Viviani we know that all fine compactified Jacobians of the same nodal curve share the same cohomology as a vector space with Hodge structures. However, the result does not say anything about whether the ring structure is independent of the choice of compactification. A recent breakthrough by Bae-Maulik-Shende-Yin [2509.05577] The intrinsic cohomology ring of the universal compactified Jacobian over the moduli space of stable curves shows that the analogue problem over the moduli spaces of curves is dependent of the compactification, and they single out a part of the cohomology, which they call 'intrinsic', where the cup product stays the same. This project is about studying an analogue of this for the case of a single curve (where some heuristic suggests that this may exhaust the whole ring, thus the ring structure may be independent of the compactification). Hands-on calculation are possible, as the first example of non-isomorphic fine compactified Jacobians already exist for rather simple nodal curves of genus 2 [1406.2299] Fine compactified Jacobians of reduced curves, where the compactified Jacobians are constructed as explicit gluing of toric surfaces.
3. Picard groups of compactified universal Jacobians
The Picard group of compactified universal Jacobians over the moduli stack Mgbar of stable curves of genus g was first described by Melo-Viviani in https://arxiv.org/abs/1007.4519. For the analogue question over the moduli stack Mgnbar of stable n-pointed curves of genus g, the literature is not complete. Kass-Pagani identified in https://arxiv.org/abs/1507.03564 a generating set for the Picard group, without giving the relations. On the other hand, Fringuelli-Viviani, in the context of more general results, gave explicit generators and relations for the Picard group of the universal Jacobian over Mgn (smooth curves only). This project is about completing the picture: giving complete descriptions (following the two different approaches above) of the Picard group of all universal compactified Jacobians over Mbargn with generators and relations, not only in the fine case, comparing the two natural generating sets, and studying wall-crossing of the generators.
Prof Aleksandr Pukhlikov
1. Birational geometry of higher-dimensional algebraic varieties
2. Birationally rigid higher-dimensional Fano varieties
3. The rationality problem for singular Fano varieties
The aim of the project is to apply the method to new classes of higher-dimensional Fano varieties and fibre spaces with singularities and solve the rationality problem for those classes of varieties, proving their birational rigidity. For this project, some acquaintance with the basics of Algebraic Geometry is needed.
Dr Alice Rizzardo
1. Examples of triangulated categories without a model
Examples of triangulated categories without a model are few and far between. Most of them use some very special property of the objects involved. The first example over a field (of characteristic zero) was constructed by myself and Van den Bergh in 2018. This PhD project concerns constructing more examples of triangulated categories over a field that do not admit a DG/topological enhancement. In particular, we will aim to construct a more geometric example.
2. Triangulated functors
This project is about characterizing triangulated functors between derived categories of two varieties. By a result of myself and Van den Bergh, any A_n functor with n at least 5 between two enhancements of the derived categories induces a triangulated functor on the level of the derived categories. The purpose of this project is to characterize triangulated functors as given by A_n functors, for some n. This is the first step to characterizing triangulated functors that admit a DG lift.