# Enumeration and moduli

A conference in algebraic geometry on the occasion of Geir Ellingsrud’s 70th birthday

The talks will start on Thursday December 6th at 11am in **Auditorium 2**, in the **Vilhelm Bjerknes building** at the University of Oslo. See the following link for details how to find this room: https://www.uio.no/om/finn-

Each day, there will be a small lunch served in the 12th floor of Niels Henrik Abels hus, which is the building next to the Vilhelm Bjerknes building.

We will also have a conference dinner on Friday December 7th at 20:30pm at Vaaghals Restaurant.

### Program

**Thursday 6th**

11:00-11:10 | Opening |

11:10-12:10 | Peskine |

12:20-13:20 | Lunch: Fredrikke building |

13:30-14:30 |
Decker |

14:45-15:45 |
Schreyer |

**Friday 7th**

10:00-11:00 |
Voisin |

11:00-11:30 | Coffee break |

11:30-12:30 |
Beauville |

12:30-13:30 | Lunch: N. H. Abel building 12th floor |

13:30-14:30 |
Lehn |

14:45-15:45 | Huybrechts |

Conference dinner at 20:30 at Vaaghals Restaurant.

**Saturday** 8**th**

10:00-11:00 |
Hulek |

11:15-12:15 |
Gulbrandsen |

12:30-13:30 |
Göttsche |

13:30-14:30 | Lunch: N. H. Abel building 12th floor |

### Titles and Abstracts:

**Arnaud Beauville**

*Limits of the trivial bundle on a curve*

Abstract: Given a family of vector bundles \(E_t\) on a curve \(C\) , such that \(E_t\) is trivial for \(t \neq 0\), what are the possible bundles \(E_0\)? I will give a complete answer in rank 2 when C is general or hyperelliptic; in that case all limit bundles are decomposable. But for \(g\geq 4\) I will exhibit a family of curves of genus g, of codimension 1 in the moduli space, carrying an indecomposable limit bundle.

**Wolfram Decker**

*A Homological Approach to Numerical Godeaux Surfaces*

Numerical Godeaux surfaces provide the first case in the geography of minimal surfaces of general type. By work of Miyaoka and Reid it is known that the torsion group of such a surface is cyclic of order at most 5, a full classification has been given for the cases where this order is 3,4, or 5. In my talk, I will discuss recent progress by Isabel Stenger towards the classification of numerical Godeaux surfaces with a trivial torsion group. Following a suggestion by Frank-Olaf Schreyer, the starting point of Stenger's work is a syzygy-type approach to the study of the canonical ring of such a surface. Particular attention is paid to the hyperelliptic curves arising in the fibration induced by the bicanonical system.

**Lothar Göttsche**

*(Refined) Verlinde formulas for algebraic surfaces.*

The celebrated Verlinde formula computes the dimension of the spaces of sections of the line bundles on moduli spaces of vector bundles on curves.

We study the analogous question of computing the holomorphic Euler characteristics of determinant line bundles on moduli spaces of vector bundles of rank 1 and 2 on an algebraic surface S, and refinements of this question. In the rank 1 case we are dealing with Hilbert schemes of points on S. In this case a partial answer, which is complete e.g. for K3 surfaces, was given many years ago by Ellingsrud, Goettsche and Lehn. We refine these results, by proving a generating formula for the elliptic genus with values in a line bundle,

interpolating between the celebrated DMVV-formula for the elliptic genus and the formulas for the holomorphic Euler characteristics of line bundles.

Then we conjecturally extend these results to moduli spaces of rank 2 sheaves on S, giving an analogue of the results on the holomorphic Euler characteristics, and refining them to the chi_y genus with values in a line bundle. These conjectures can further be extended to moduli spaces of Higgs sheaves on S.

**Martin Gulbrandsen**

*Stabilizer groups of sheaves on abelian threefolds*

Recent developments in generalized Donaldson--Thomas invariants for

abelian threefolds raise concrete questions on the stabilizer group of a

coherent sheaf with respect to the actions by translations and twists

with homogeneous line bundles. In particular we attempt to understand

what we think of as partially semi-homogeneous sheaves, i.e. sheaves

with stabilizer group of small, but positive dimension. This is a status

report on ongoing work together with Riccardo Moschetti.

**Klaus Hulek**

*Moduli Spaces of Cubic Threefolds — Geometry and Topology*

Cubic threefolds were the first class of varieties who were shown to be unirational but not rational (Clemens,Griffiths). The key tool of the proof is the intermediate Jacobian, a principally polarized abelian variety of dimension 5. There is a second link to Hodge theory, namely via cubic fourfolds (Allcock, Carlson, Toledo) which leads to a $10$-dimensional ball quotient model. Looking at cubic threefolds from these different points of view leads to various geometrically relevant compactifications of the moduli space of cubic threefolds. In this talk I will discuss the geometry and the topology of these spaces. This is joint work with S. Casalaina-Martin, S. Grushevsky and R. Laza.

**Daniel Huybrechts**

*K3 surfaces/categories versus cubic fourfolds and Hilbert schemes*

In this talk I shall survey results on the Hassett-Kuznetsov correspondence between K3 surfaces and cubic fourfolds with special emphasis on Hilbert schemes.

**Manfred Lehn**

*On the involution on the symplectic 8-fold associated to a cubic
fourfold*

The moduli space of generalised twisted cubics on a smooth cubic fourfold Y admits a contraction to an irreducible symplectic 8-fold Z. The manifold Z carries an anti-symplectic involution. I would like to discuss geometric questions arising in connection with this involution.

**Christian Peskine**

*The projective ubiquity of the Jacobian of a genus 2 curve. Embeddings, tangencies, ramication, virtual hyperplanes. A nostalgic walk.*

**Frank-Olaf Schreyer**

*Curves of genus 11 with several pencils of degree 6. *

Green's conjecture says that vanishing syzygies of a canonical curve is

equivalent to the non-existence of certain linear series on the curve.

Turning things around, we might hope that many syzygies imply the existence

of many linear systems.

In this talk I will report on work of Hanieh Keneshlou, who used this approach to study

the scheme of curves of genus 11 with several pencils of degree 6.

**Claire Voisin**

*Segre numbers of tautological bundles on Hilbert schemes of surfaces*

Abstract: The top Segre classes of tautological bundles of punctual Hilbert schemes of surfaces are numbers which can be assembled in a generating function. The Lehn conjecture gives a complete description of this function, as obtained from a rational function by a change of variables.