**Abstract**
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Gavin BROWN and Miles REID

**Title:** *Tom and Jerry and Sarkisov links, I and II*

**Abstract:** We construct Fano 3-folds anticanonically embedded in weighted
projective space in codimension 4 using Kustin-Miller unprojection
(Papadakis-Reid unprojection of Type I). This constructs those Fano 3-folds
that we expect to admit Type I projections in at least two ways (Tom and Jerry),
and extending the corresponding Sarkisov links distinguishes the cases according
to the Mori fibre spaces they link to.

Alvaro Nolla de Celis

**Title:** *Dihedral groups and $G$-Hilb*

**Abstract:** The special McKay correspondence states that the minimal
resolution $Y$ of the singular quotient $C^2/G$ by a finite subgroup
$G$ in $GL(2,C)$ is the $G$-invariant Hilbert scheme $G\text{-Hilb}$.
In this talk I will consider this correspondence when $G$ is a binary
dihedral group in $GL(2,C)$, and explain how we can give an explicit
description of $G\text{-Hilb}$ via $G$-graphs and the moduli space
$M(Q,R)$ of stable representations of the McKay quiver.

Osamu FUJINO

**Title:** *On log canonical flops*

**Abstract:** We explain Koll\'ar's example of a log canonical
flopping contraction. It does not have a flop.

Klaus HULEK

**Title:** *Calculating the Mordell-Weil rank of elliptic threefolds*

**Abstract:** In this talk I will discuss a method to calculate the Mordell-Weil
rank of an elliptic threefold. The main idea is to reduce this problem to
calculating the cohomology of a singular hypersurface in a weighted
projective $4$-space. I then describe a method for calculating the
cohomology of a certain class of singular hypersurface.
As an application I discuss a class of elliptically fibered Calabi-Yau
threefolds over del Pezzo surfaces which were first constructed by
Hirzebruch. This is joint work with R. Kloosterman.

Akira ISHII

**Title:** *Dimer models and the special McKay correspondence*

**Abstract:** A dimer model is a bi-colored graph on a $2$-torus,
which encodes the information of a quiver with relations.
A typical example is given by a tessellation of regular hexagons,
in which case the associated quiver is the McKay quiver for a finite
abelian subgroup of $SL(3, C)$.
When a dimer model is \lq non-degenerate', the moduli space of
representations of the associated quiver with relations (with dimension
vector $(1,1, ... ,1)$ and with respect to a generic stability
parameter) is a crepant resolution of the $3$-dimensional Gorenstein
affine toric variety associated with a lattice polygon, which is
combinatorially determined by the dimer model.
Fundamental questions on dimer models are:

1. Describe \lq consistency condition' on a dimer model,
which ensures that the moduli space is derived equivalent to the path algebra of
the associated quiver with relations.

2. For an arbitrary $3$-dimensional Gorenstein affine toric
variety, construct dimer models which satisfies the condition in Question 1.

In this talk, I will explain the behavior of the dimer
models under the removal of a vertex from the toric diagram (and taking the convex
hull of the rest), and how this can be applied to the above two questions.
The triangle case is crucial and depends on computations on the \lq special
representations' of Wunram-Riemenschneider for two-dimensional cyclic quotient singularities. This is a joint work with Kazushi Ueda.

Yukari ITO

**Title:** *Existence of crepant resolutions*

**Abstract:** There are several kinds of generalizations of the McKay correspondence,
but all of them need a crepant resolution. It is known that there exist crepant
resolutions in dimension 2 or 3 for any Gorenstein quotient singularities, but
it is not the same in higher dimension. I would like to talk on existence of
higher dimensional crepant resolutions.

Anne-Sophie KALOGHIROS

**Title:** *The defect of Fano $3$-folds*

**Abstract:** Let $X$ be a quartic $3$-fold in $P^4$ with no worse than terminal
singularities. The Grothendieck-Lefschetz theorem states that the
Picard rank of $X$ is $1$, i.e. that every Cartier divisor on $X$ is a
hyperplane section of $X$. However, no such result holds for the group
of Weil divisors of $X$ if $X$ is not factorial.
In this lecture, I will give a bound for the rank of the group of Weil
divisors of $X$ when $X$ is a Fano $3$-fold with terminal Gorenstein
singularities. This bound is optimal in the case of the quartic $3$-fold.
This bound is obtained using birational geometry.
I will also show how these methods yield a "classification"
of non factorial terminal quartic $3$-folds.

Yongnam LEE

**Title:** *Log minimal model program for the moduli space of stable curves
of low genus*

**Abstract:** In this talk, we completely describe the log minimal model
program for the moduli space of curves of stable curves of genus $\le 3$,
and the second small contraction of genus $4$.
It is jointed work with Donghoon Hyeon.
The log minimal model program for the moduli space of curves of stable curves
of genus $g$ was introduced by Brendan Hassett.
The first divisorial contraction, small contraction,
and flip were worked out recently by Hassett and Hyeon.

Timothy LOGVINENKO

**Title:** *A derived approach to the geometric McKay correspondence*

**Abstract:** We describe a three dimensional generalization of the geometric
McKay correspondence constructed by Gonzales-Sprinberg and Verdier
in dimension two. More precisely, we show that the Bridgeland-King-Reid
derived category equivalence induces a natural geometric correspondence
between irreducible representations of $G \subset SL_3(C)$ and
closed subschemes of the exceptional set of $G\text{-Hilb}(C^3)$. Applied in
dimension two this produces precisely the classical McKay.
We further show that for an abelian G this correspondence can be
calculated explicitly and that the result is intrinsically related to
the toric combinatorics of the 'Reid's recipe'.

James McKernan

**Title:** *Batyrev's conjecture*

**Abstract:** I describe some work of Brian Lehmann who proves some results about
the cone of nef curves, analagous to the cone theorem and contraction theorem,
which concern the $K_X$-negative side of the cone of effective curves.

Yoshinori NAMIKAWA

**Title:** *Induced nilpotent orbits and birational geometry*

**Abstract:** Nilpotent orbit closures in a complex simple Lie algebra
$g$ have symplectic singularities. In this lecture, I will pose
a conjecture that any $Q$-factorial terminalization of them is
obtained as a generalized Springer map, and will prove that
the conjecture actually holds true when $g$ is a classical Lie algebra.

Keiji OGUISO

**Title:** *Mordell-Weil groups of abelian fibered hyperkaehler manifolds and O'Grady's $10$-dimensional examples*

**Abstract:** I would like to talk about Mordell-Weil rank of an abelian
fibered hyperkaehler manifold. First, I notice that the maximal
possible Mordell-Weil rank is $20$ among all currently known hyperkeahler
manifolds. Then, I would like to explain how one can obtain examples having
the maximal Mordell-Weil rank $20$, by using deformation and explicit
birational geometry of O'Grady's $10$-dimensional examples.

Jongil PARK

**Title:** *A new family of algebraic surfaces with $p_g = 0$ and $K^2 > 0$*

**Abstract:** One of the fundamental problems in the classification of complex
surfaces is to find a new family of simply connected surfaces with $p_g = 0$ and
$K^2 > 0$. Although a large number of non-simply connected complex surfaces of
general type with $p_g = 0$ and $K^2 > 0$ have been known,
until recently the only previously known simply
connected, minimal, surface of general type with $p_g = 0$ and $K^2 > 0$
was Barlow surface which has $K^2 = 1$.
The natural question arises if there are other simply
connected surfaces of general type with $p_g = 0$ and $K^2 > 0$
except Barlow surface.
In 2004 I constructed a new family of simply connected symplectic $4$-manifolds
with $b_2^+ = 1$ and $1 \le K^2 \le 2$ using a rational blow-down surgery.
After this construction, it had been an intriguing question
whether these symplectic $4$-manifolds admit a complex structure.
In 2006 Yongnam Lee and myself constructed a new family of simply connected,
minimal, complex surfaces of general type with $p_g = 0$ and $1 \le K^2 \le 2$
by modifying the symplectic $4$-manifolds constructed above.
Our main techniques used here are a
rational blow-down surgery and $Q$-Gorenstein smoothing theory.
In 2007 Heesang Park , Dongsoo Shin and myself successfully found a right
configuration to produce simply connected, minimal, complex surfaces of general type
with $p_g = 0$ and $3 \le K^2 \le 4$ using the same technique as above.
In this talk, I will sketch how to construct such a new family of simply
connected surfaces of general type using a rational blow-down surgery and
$Q$-Gorenstein smoothing theory.
If time allows, I will also present a new family of surfaces of general type
with $p_g = 0$ and small homology group.
This talk is based on joint works with Yongnam Lee and with Heesang Park and
Dongsoo Shin.

Thomas PETERNELL

**Title:** *Varieties with generically nef tangent bundles*

**Abstract:** I will discuss the notion of generically nef line bundles and
vector bundles and basic properties.
The main theme of the talk is to discuss which manifolds have
generically nef tangent bundles and to indicate applications to
classification theory.

Miles REID

**Title:** *Big key varieties for surfaces with $p_g = 1$, $K^2 = 2$ (according to Stephen Coughlan)*

**Abstract:**

Hiromichi TAKAGI

**Title:** *$Q$-Fano $3$-folds and varieties of power sums*

**Abstract:** We discuss some examples of $Q$-Fano $3$-folds
which can be described by varieties of power sums.

Yukinobu TODA

**Title:** *Stability conditions and Donaldson-Thomas type invariants*

**Abstract:** Donaldson-Thomas invariant is a counting invariant of curves on
Calabi-Yau 3-folds. Now there are several expected properties of generating functions
of DT-invariants, say DT-PT correspondence, rationality conjecure, DT-NCDT
correspondence, and flop invariance. In this talk, I will show that all the
above properties can be deduced from the wall-crossing formula in a certain
triangulated subcategory of the derived category of coherent sheaves.

Michael WEMYSS

**Title:** *McKay Correspondence for two dimensional rational surface singularities*

**Abstract:** I will describe a certain correspondence for all two dimensional
quotient singularities, and will explain how some aspects generalize
to all rational two dimensional surface singularities.