# 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

Vladimir Guletskii offers the following range of new ideas towards deformation of rational equivalence of 0-cycles on algebraic varieties in families over a field. Let S be a nice base scheme, say a smooth algebraic curve or the spectrum of a DVR. Let X be a scheme, smooth and projective over S. Consider the corresponding Nisnevich sheaf Z_0(X/S) of relative dimension 0 algebraic cycles on the category of Noetherian schemes over S arising from the scheme X over S, as suggested by Suslin and Voevodsky in their paper "Relative cycles and Chow sheaves" (a different but equivalent approach to sheaves of relative algebraic cycles is given in Kollar's book "Rational curves on algebraic varieties"). Etale morphisms from schemes over S to the sheaf Z_0(X/S) allows us to consider the latter as a Nisnevich site with a sheaf of regular functions on it, similar to what we do in stacks theory (see the Stacks Project online or Olsson's book). Following Illusie's "Complexe Cotangent et Deformations" one can also construct the sheaf of Kaehler differentials and the tangent sheaf on the ringed site Z_0(X/S). This all allows us to put a trustworthy theoretical basis under the heuristic approach presented in the Green-Griffiths book "On the tangent space to the space of algebraic cycles". In particular, we can now consider regular morphisms from the projective line P^1 to Z_0(X/S) and thus speak about deformation of rational curves on the space Z_0(X/S), following Kollar but now working with rational curves on the space of 0-cycles Z_0(X/S). These visionary ideas give birth to 3 - 5 research projects for PhD students, and Guletskii's group in Liverpool is currently seeking energetic and ambitious PhD students to place them at the forefront of this vibrant research. More details are given on Vladimir's webpage: http://pcwww.liv.ac.uk/~guletski/

**Dr Nicola Pagani**

**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: Towards an enumerative theory of curves inside moduli spaces.**

Curves in moduli spaces (of curves) have been very well studied in the context of fibered surfaces. Nowadays, there are modern theories (for example, Gromov-Witten, but there are many others) that predict, and in some cases enumerate, curves in a given target variety X. In this project, we want to think of this target variety X as a moduli space of curves itself, and see how many (rational, higher genus?) curves are predicted by these theories, and if these predictions are attained. Starting calculations would be on \bar{M}_{0,6}, \bar{M}_{1,2}, and \bar{M}_2.

**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).

The above **three projects** focus on moduli spaces of curves, their cohomological and enumerative aspects, and especially the interplay with Gromov-Witten theory. Projects along these fields or topics concerning the tautological rings, on moduli spaces of coverings (Hurwitz spaces), and on various moduli spaces of Jacobians can be designed. To undertake the proposed PhD projects, candidates are required to have familiarity with algebraic curves and/or Riemann surfaces. A general knowledge of algebraic geometry and topology is desirable, but can be also developed in itinere.

### 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. Families of on-Fourier-Mukai functors**

Almost all the functors we know between the bounded derived categories of two smooth projective varieties can be expressed as Fourier-Mukai functors, and therefore can be lifted to functors between differential graded (DG) categories. The first example of a functor that doesn't admit a DG-lift was produced by myself and Prof. Van den Bergh a couple of years ago. It is still the only known example in characteristic zero.

The methods used in the construction of our example, as well as general considerations, indicate that there should be many more of these "pathological" functors. This project aims at constructing new examples, and even better, families of non-Fourier-Mukai functors. The project will use ideas from our original paper as well as very recent developments in the field. These new examples will give us important insight in the behavior of derived categories and how they compare to the world of DG categories.

**2. Crepant resolutions of quotient singularities **

Let G be a finite subgroup of SL(3,C). The crepant resolutions of the quotient singularity C^3/G have been classified in 2004 by Craw and Ishii when G is an abelian group. When G is not abelian, this classification becomes much harder for several technical reasons, though it is known in some cases thanks to work of Wemyss. The aim of this project is to compute some specific examples to shed light on the non-abelian case. This will involve using computer algebra systems such as Singular and Magma.