Workshop
Derived Categories in Algebraic Geometry
2325 May 2012
Université de Bourgogne ~ DIJON
Program
The workshop will be centered around the following two minicourses

Geometry and derived categories of cubic 3folds and 4folds
by E. MACRI & P. STELLARI
Abstract
The aim of these lectures is to recall some basic properties and classical
results concerning the geometry of smooth cubic hypersurfaces of dimension
3 and 4. We explain a way to translate them in the language of derived
categories of coherent sheaves and outline how this approach helps in
solving geometric questions.

Introduction to Homological Projective Duality (after Kuznetsov)
by L. MANIVEL & R. ABUAF
Abstract
These lectures will be an introduction to a circle of ideas due
to A. Kuznetsov, aiming at understanding classical projective
duality at the level of derived categories. We will discuss
several examples, showing how homological projective duality
can be a powerful tool for describing the derived categories
of certain Fano or CalabiYau varieties, obtained as linear
sections of homogeneous spaces.
and a series of research talks :

Comparing Hassett's and Kuznetsov's Special Cubic Fourfolds
Nicolas ADDINGTON
Abstract
Hassett described a countable set of NoetherLefschetz divisors in the moduli space of cubic fourfolds,
expecting a cubic to be rational if and only if it lies in one of these divisors.
Kuznetsov described a K3 category associated to a cubic and conjectured the cubic is rational if and only
if this category is the derived category of an honest, geometric K3 surface.
In joint work with Richard Thomas, we can show that if the K3 category is geometric then the cubic lies
in one of Hassett's divisors, and for cubics in a Zariski open subset of each divisor,
the K3 category is geometric. (We are still working on showing that the subset is closed.)
In particular we can show that for a very general cubic, the K3 category has no pointlike objects.

Determinantal varieties and HPD
Marcello BERNARDARA
Abstract
A.Bondal proposed the following problem: do the derived category of any smooth projective variety embed fully
faithfully into the derived category of a Fano variety?
In this talk I will present a joint work with M.Bolognesi and D.Faenzi,
describing the homological projective duality between Segre and (categorical resolutions of)
determinantal varieties. As a consequence, we get a list of varieties with a positive answer
to Bondal's question. As an application, the categorical resolution of singularities of determinantal
cubic 3folds and 4folds can be explicitely described.

Rational cubic fourfolds containting a plane with nontrivial Clifford invariant
Michele BOLOGNESI
Abstract
In this talk I will showcase a general class of smooth rational cubic fourfolds X containing a plane
whose associated quadric surface bundle does not have a rational section. Equivalently, the Brauer
class B of the even Clifford algebra over the discriminant cover (a K3 surface S of degree 2)
associated to the quadric bundle, is nontrivial. These fourfolds provide nontrivial examples verifying
Kuznetsov's conjecture on the rationality of cubic fourfolds containing a plane. Indeed, using
homological projective duality for grassmannians, one obtains another K3 surface S' of degree 14
and a nontrivial twisted derived equivalence A_{X} = D^{b}(S;B) = D^{b}(S'), where A_{X} is
Kuznetsov's residual category associated to the cubic hypersurface X.

On the derived category of the Cayley plane
Daniele FAENZI
Abstract
Many rational homogeneous varieties admit full strongly exceptional collection (Beilinson, Kapranov, Orlov, Kuznetsov, Fonarev),
and even Lefschetz collections consisting of homogeneous bundles. Although this conjecturally holds for all such varieties,
still many cases are missing. I will show a construction of such a collection for the Cayley plane,
the minimal orbit of the exceptional group E_{6}. The method involves restriction to quadrics in the Cayley plane,
combined with a rephrasing of an idea of BondalOrlov.
If time allows, I will present some hints on how to construct Lefschetz collections on some
other homogeneous varieties. This work is in collaboration with Laurent Manivel.
Participants
 Paolo STELLARI (Milano)
paolo.stellari@unimi.it
 Emanuele MACRI (Ohio State University)
macri.6@math.osu.edu
 Laurent MANIVEL (Grenoble)
Laurent.Manivel@ujfgrenoble.fr
 Roland ABUAF (Grenoble)
Roland.Abuaf@ujfgrenoble.fr
 Chris PETERS (Grenoble)
chris.peters@ujfgrenoble.fr
 Nicolas ADDINGTON (Imperial College London)
n.addington@imperial.ac.uk
 Agnieszka BODZENTASKIBINSKA (Varsovie)
a.bodzenta@mimuw.edu.pl
 Ada BORALEVI (SISSA Trieste)
ada.boralevi@sissa.it
 Hathurusinghege Dulip Bandara PIYARATNE (Edinburgh)
H.D.B.Piyaratne@sms.ed.ac.uk
 Magnus ENGENHORST (Freiburg)
magnus.engenhorst@math.unifreiburg.de
 Mateusz MICHALEK (Varsovie)
ajcha2@poczta.onet.pl
 Martina ROVELLI (Genoa)
martina_rovelli@yahoo.it
 Zhi JIANG (Orsay)
Zhi.Jiang@math.upsud.fr
 David FINSTON (New Mexico State University)
dfinston@nmsu.edu
 Marcello BERNARDARA (Toulouse)
mbernard@math.univtoulouse.fr
 Michele BOLOGNESI (Rennes)
michele.bolognesi@univrennes1.fr
 Daniele FAENZI (Pau)
daniele.faenzi@univpau.fr
 Alexander SAMOKHIN (Mayence)
alexander.samokhin@gmail.com
 Francesco AMODEO (Milano)
francescoamodeo87@gmail.com
 Robert LATERVEER (Strasbourg)
robert.laterveer@math.unistra.fr

Marko ROCZEN (Berlin)
roczen@mathematik.huberl
 Jan NAGEL (Dijon)
 Johann BOUALI (Dijon)
 Romain BOILLAUD (Dijon)
 Lucy MOSERJAUSLIN (Dijon)
 Adrien DUBOULOZ (Dijon)
 Charlie PETITJEAN (Dijon)
 Bachar AL HAJJAR (Dijon)
 Shameek PAUL (Dijon)
Schedule
Wednesday 
Thursday 
Friday 

MANIVEL/ABUAF I 10:0011:00 
MANIVEL/ABUAF III 10:0011:00 
MACRI II 11:1512:15 
FAENZI 11:1512:15 
Lunch Break 12:3014:00 
MACRI I 14:0015:00 
STELLARI II 14:0015:00 
ADDINGTON 14:0015:00 
STELLARI I 15:1516:15 
BOLOGNESI 15:1516:15 

Coffee Break 
BERNARDARA 16:4517:45 
MANIVEL/ABUAF II 16:4517:45 

19:30 Dinner 
The time table and the abstracts are aslo available as a
pdf file .
Suggested Reading
Books, survey papers:
 1. D. Huybrechts, FourierMukai transforms in algebraic geometry,
Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford 2006.
 2. Andrei Caldararu, Derived categories of sheaves: a skimming,
http://arxiv.org/abs/math/0501094
Research Papers:
 1. Emanuele Macri, Paolo Stellari, Fano varieties of cubic fourfolds containing a plane
http://arxiv.org/abs/0909.2725
 2. Marcello Bernardara, Emanuele Macri, Sukhendu Mehrotra, Paolo Stellari, A categorical invariant for cubic threefolds,
http://arxiv.org/abs/0903.4414
 3. Alexander Kuznetsov, Homological Projective Duality,
http://arxiv.org/abs/math/0507292
 4. Alexander Kuznetsov, Derived categories of cubic and V14 threefolds,
http://arxiv.org/abs/math/0303037
 5. Alexander Kuznetsov, Derived categories of cubic fourfolds,
http://arxiv.org/abs/0808.3351
 6. Alexander Kuznetsov, Hyperplane sections and derived categories,
http://arxiv.org/abs/math/0503700
Practical Informations
Access
The workshop will take place at the
Mathematical Institute of Burgundy (IMB) in Dijon.
More information, including a map of the campus, on how to reach us can be found
here .
You can also consult
this detailed map to find your way to the Math. building form the bus stop
Marechal.
Accomodation
Please consult
this list to find out in which hotel your are accomodated.
Contacts
For more information, please contact:
Johannes NAGEL
Caroline GERIN
Our sponsors