# Sample PhD Projects in Algebraic Geometry

## 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/

**Dr Nicola Pagani**

The three projects below focus on moduli spaces of curves, related enumerative aspects, and especially the interplay with Gromov-Witten theory. To undertake one of these directions, candidates are required to have familiarity with the basics of algebraic geometry, the example of algebraic curves and/or Riemann surfaces, and to have some familiarity with the cohomology ring.

**Project 1: Algebro-geometric perspective on moduli spaces of meromorphic differentials**

The moduli space of differentials H_g parametrizes pairs (C, \omega), with C a smooth Riemann surface of genus g and \omega a meromorphic differential on C. This moduli space has been classically studied from the perspective of holomorphic dynamics. Recently, these moduli spaces have come to the attention of algebraic geometers studying moduli spaces of curves. The moduli space H_g admits a stratification determined by prescribing the multiplicities of zeroes and poles of the meromorphic differential, a partition (k1, ..., kn) of 2g-2. Farkas and Pandharipande have given a modular compactification of the image of these strata in Mgn. In the same paper, the two authors observe that a systematic algebro-geometric study of the stata has so-far been neglected.

This project will study the estrinsic algebro-geometric properties of these strata. The main proposal is to compute the cohomology classes of these strata in terms of tautological classes, starting from some low-genus examples. Some working understanding of intersection theory and of limit linear series will be part of the techincal background developed during the development of this PhD project.

**Project 2: The geometry and combinatorics of compactified universal Jacobians**

We propose to study the geometry of the compactification of the universal Jacobian, the moduli space that parametrises pairs (C,L) where C is a smooth algebraic curve of given genus and L is a line bundle of some fixed degree on C. A classical modular compactification of the moduli space of curves is given in terms of stable curves. There are several different modular compactifications that admit a map to the moduli space of stable curves. The set of all such compactifications modulo isomorphisms has been recently studied in a series of works by Kass and Pagani. For the Jacobian of a single stable curve, we know the following two things. There are two natural (and independent) questions.

1) Recent work by Migliorini-Shende-Viviani shows that the cohomology of two compactified Jacobians of the same stable curve are isomorphic as vector spaces. Here we want to ask if the same is true for two compactified universal Jacobians (over the same moduli space). This project will first require to understand the details of the proof of Migliorini-Shende-Viviani's result, which requires developing some understanding of intersection cohomology and of the classical decomposition theorem by Beilinson-Bernstein-Deligne.

2) How many non-isomorphic compactified Jacobians exist for a fixed genus? By the work of Kass-Pagani this question can be reformulated as the combinatorial problem of counting the number of chambers of a certain hyperplane arrangement on a real torus, modulo the action of a certain group. The theory to count the chambers of a hyperplane arrangement in a vector space is classical and due to Zavlasky, and recent results extend that theory to the case of arrangements on a torus.

**Project 3: Computations in the tautological ring of the moduli spaces of curves**

This is a project for someone with a strong background in algorithms and programming OR, alternatively, for someone who (besides algebraic geometry), very much likes playing with power series. The cohomology of the moduli spaces of curves plays a role in several fields, and has a nice, albeit complicated, combinatorial structure. The cohomology classes that play a role in other theories, are often contained in a subring called tautological ring. Very recently, Pandharipande, Pixton, and Zvonkine have found a big set of relations in the tautological ring, that comprises all the known ones, and that conjecturally exhaustes the whole set of relations. Petersen-Tommasi have shown that this ring is not a Poincarè duality ring: there are tautological classes that intersect to zero with all tautological classes of the complement codimension. However, it still makes sense to study all the so-called Gorenstein relations. In this project, we want to do computer-assisted calculations and try to:

- Give a unified combinatorial description of the set of Gorenstein relations,
- Find new Gorenstein relations (not in the Pandharipande-Pixton-Zvonkine set), and decide if they are actual relations (thus disproving the conjecture), or finding other examples of tautological rings that are not Gorenstein. (Note: this topic of research is very vibrant: several top researchers worldwide work on related issues, so this might very well be changed in due course).

### 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.