# Schedule of the Workshop "Geometry of Complex Threefolds"

## Monday, March 10

 09:30 - 10:30 Meng Chen: On the geography of 3-folds of general type I 10:30 - 11:00 Coffee break 11:00 - 12:00 Alexandru Dimca: Betti numbers of hypersurfaces and defects of linear systems I 12:00 - 14:00 Lunch break 14:00 - 15:00 Tobias Dorsch: Global deformations of quartic double solids 15:00 - 16:00 Talk or discussion or free time 16:00 - 16:30 Tea and cake

## Tuesday, March 11

 09:30 - 10:30 Alexandru Dimca: Betti numbers of hypersurfaces and defects of linear systems II 10:30 - 11:00 Coffee break 11:00 - 12:00 Meng Chen: On the geography of 3-folds of general type II 12:00 - 14:00 Lunch break 14:00 - 15:00 Miles Reid: Graded rings and Fano 3-folds 15:00 - 16:00 Talk or discussion or free time 16:00 - 16:30 Tea and cake

## Wednesday, March 12

 09:30 - 10:30 Meng Chen: On the geography of 3-folds of general type III 10:30 - 11:00 Coffee break 11:00 - 12:00 Alexandru Dimca: Betti numbers of hypersurfaces and defects of linear systems III 12:00 - 14:00 Lunch break 14:00 - 15:00 Yongnam Lee: Q-Gorenstein deformations and their applications 15:00 - 16:00 Talk or discussion or free time 16:00 - 16:30 Tea and cake

## Thursday, March 13

 09:00 - 10:00 Alexandru Dimca: Betti numbers of hypersurfaces and defects of linear systems IV 10:00 - 10:30 Coffee break 10:30 - 11:30 Meng Chen: On the geography of 3-folds of general type IV 11:30 - 14:00 Lunch break 14:00 - 15:00 Michael McQuillan: 3-folds foliated in curves 15:00 - 16:00 Talk or discussion or free time 16:00 - 16:30 Tea and cake

# Abstracts

(Underlined titles can be clicked for the video recording)

#### Meng Chen: On the geography of 3-folds of general type

In this series of lectures, I will briefly introduce some results concerning the geometry inspired by the pluricanonical system $|mK|$ of threefolds of general type. I will talk about the general method to estimate the lower bound of the canonical volume $K^3$ and the proof of a 3-dimensional Noether's inequality. In the last part, I will discuss some new examples of canonically fibered varieties of general type. The titles of the four lectures are as follows:

1. A review on the geometry of $|mK|$

2. Lower bound for the volume $K^3$

3. A Noether type of inequality and its variant

4. Some new examples of canonically fibered varieties of general type.

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#### Alexandru Dimca: Betti numbers of hypersurfaces and defects of linear systems

Our approach is a generalization of Griffiths' results expressing the cohomology ofa smooth hypersurface $V: f=0$ in a projective space $\mathbb{P}^n$ in terms of some graded pieces of the Jacobian algebra of $f$.

We will start by recalling these classical results.

Then we explain that when the hypersurface $V$ becomes singular, there is a natural spectral sequence, starting with some graded pieces of the Koszul cohomology groups of $f$ and converging to the graded pieces of the cohomology of the complement $U= \mathbb{P}^n \setminus V$ with respect to the order pole filtration.

When $V$ has only isolated singularities, the Koszul cohomology is given by the syzygies involving the partial derivatives of $f$, and their number is related to the defect of some linear systems. When the singularities of $V$ are nodes or, more generally, weighted homogeneous singularities, our results can be made very precise. Finally we'll discuss the case of nodal 3-folds in $\mathbb{P}^4$ and the topology of their small resolutions.

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#### Tobias Dorsch: Global deformations of quartic double solids.

This talk is concerned with a special case of the question of whether global deformations (in the category of compact complex manifods) of Fano manifolds with Picard number 1 are also Fano. After briefly reviewing some known results I want to give some ideas of why this question has a positve answer for global deformations of quartic double solids, i.e. double covers of $P^3$ branched along smooth quartic hypersurfaces.

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#### Yongnam Lee: Q-Gorenstein deformations and their applications.

In this talk we will discuss Q-Gorenstein schemes and Q-Gorenstein morphisms in a general setting. Based on the notion of Q-Gorenstein morphism, we define the notion of Q-Gorenstein deformations and discuss their properties. Versal properties of Q-Gorenstein deformations and their applications to higher dimensional varieties are also considered. This is joint work with Noboru Nakayama.

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TBA

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#### Miles Reid: Graded rings and Fano 3-folds

It is known that there are about 60000 possibilities for the Hilbert series of anticanonical rings of Q-Fano 3-folds in the Mori category. Graded ring methods allow the construction of Q-smooth Q-Fano 3-folds corresponding to many hundreds of these Hilbert series. One expects the methods to be applicable to many thousand more. There are however somesurprises, with many Hilbert series corresponding to several topologically distinct Q-smooth Q-Fano 3-folds.

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