2nd Swiss-French workshop on algebraic geometry

Jérémy Blanc - Universität Basel - Mathematik

Enney (near Gruyères, Fribourg, Switzerland), February 18-22, 2013

The workshop was held in Enney from February 18 to 22, 2013.

Mini-courses

In the morning, there were three mini-courses of 5 hours (3 times one hour each day).

Ivan ARZHANTSEV (Moscow)	Homogeneous spaces and Cox rings

Jaroslaw A. WISNIEWSKI (Warsaw)	An introduction to Mori theory

Jérémy BLANC (Basel) Adrien DUBOULOZ (Dijon)	Automorphisms of affine algebraic surfaces via birational geometry

In the afternoon, we had research talks of 50 minutes.

Schedule

Talks: In Enney, ("Centre l'Ondine, Viva-Gruyère" )

Monday February 18	Tuesday February 19	Wednesday February 20	Thursday February 21	Friday February 22
12h30 welcome	breakfast 8h45-9h45 mini-course 1 10h15-11h15 mini-course 2 11h45-12h45 mini-course 3	breakfast 8h45-9h45 mini-course 1 10h15-11h15 mini-course 2 11h45-12h45 mini-course 3	breakfast 8h45-9h45 mini-course 1 10h15-11h15 mini-course 2 11h45-12h45 mini-course 3	breakfast 8h45-9h45 mini-course 1 10h15-11h15 mini-course 2 11h45-12h45 mini-course 3
lunch	lunch	lunch	lunch
14h30-15h30 mini-course 1 16h00-17h00 mini-course 2 17h30-18h30 mini-course 3 dinner	time for discussion / enjoying the mountain 17h20-18h10 talk - I. Bazhov 18h30-19h20 talk - M. Leuenberger dinner	time for discussion / enjoying the mountain 17h20-18h10 talk - A. Calabri 18h30-19h20 talk - M. Pieropan dinner	time for discussion / enjoying the mountain 17h20-18h10 talk - G. Brown 18h30-19h20 talk - K. Palka dinner

Mini-courses - titles and abstracts

Ivan ARZHANTSEV - Homogeneous spaces and Cox rings
	The aim of this course is to introduce an important invariant of an algebraic variety - the total coordinate ring, or the Cox ring, - and to discuss how to apply it in the theory of algebraic transformation groups. We plan to study algebraic properties of Cox rings, geometric properties of the canonical quotient presentation, and to illustrate them on examples from toric geometry and the theory of embeddings of homogeneous spaces. Main applications concern classification problems and a description of the automorphism group of a complete variety with torus action. Lecture 1. Basic properties of linear algebraic groups and their homogeneous spaces. Lecture 2. Geometric Invariant Theory and variation of quotients. Lecture 3. Universal torsors and Cox rings: definitions, properties, examples. Lecture 4. Cox rings of homogeneous spaces, almost homogeneous varieties, and varieties with torus action. Lecture 5. Automorphisms of algebraic varieties: Demazure roots via Cox rings.
Jaroslaw A. WISNIEWSKI - An introduction to Mori theory
	The course will give an introduction to basics notions of higher dimensional algebraic geometry related to positivity of line bundles and Mori cone theorem. We intend to discuss the following topics: Ample and very ample line bundles, cohomology, theorems A and B of Serre. Intersection of curves and divisors, numerical equivalence, Kleiman theorem. Nef divisors, cone of curves and cone of nef divisors. Big divisors, Kodaira lemma. Stein factorisation and contractions. Fundamental triviality of the Mori program: faces of cones vs contractions. Parameter spaces for morphisms of curves, dimension estimate via Riemann-Roch. Existence of rational curves, K_X not nef, char(k)>0, Frobenius morphism trick. Rational curves on Fano manifolds, coming back to char(k)=0. The cone theorem of Mori. Contractions of extremal rays. Case of surfaces.
Jérémy BLANC, Adrien DUBOULOZ - Automorphisms of affine algebraic surfaces via birational geometry
	The aim is to give tools of birational geometry of surfaces (blow-ups and contractions) that can be used to describe affine surfaces and their automorphisms. Lecture 1. Blow-ups and contractions Lecture 2. Proof of Jung's theorem using birational geometry Lecture 3. Generalisation to other surfaces admitting completion by a zigzag, and discussion of the case with at least one self-intersection ≤ -3. Lecture 4. The case of surfaces admitting completion by a zigzag with all self-intersections ≥ -2. Lecture 5. Existence of affine surfaces with very large groups (not generated by any countable union of algebraic groups)

Talks - titles and abstracts

Ivan BAZHOV -Automorphisms of toric varieties.
	Let X be a complete toric variety and let Aut(X) be the automorphism group of X. We will give a description of Aut(X)-orbits on X. Our proof will use Cox rings technique.
Gavin BROWN - Reducible moduli of 3-folds
	A Fano 3-fold X is naturally embedded in (weighted) projective space by multiples of its anticanonical class. When that embedding is of codimension at most 3 (that is, the projective space has dimension at most 6), then good deformation families are well known, either as complete intersections, or as regular pullbacks from Grass(2,5) in the style of Mukai. In 2001, Takagi discovered three numerical cases of Fano 3-folds in codimension 4 that are realised by at least two distinct deformation families. I show how this extends to most cases in codimension 4 using the Papadakis--Reid unprojection theorem and a detailed analysis of the ways of imposing linear spaces on regular pullbacks from Grass(2,5). This is joint work with Kerber and Reid. I will also report on recent joint work with Georgiadis which shows how similar methods can be applied to find distinct deformation families of polarised Calabi-Yau 3-folds and canonical 3-folds. If it's appropriate to mention reading (since it's a workshop) the relevant paper is: Fano 3-folds in codimension 4, Tom and Jerry. Part I, with Michael Kerber and Miles Reid, Compositio Math. 148 (2012), 1171-1194
Alberto CALABRI - On plane Cremona transformations of fixed degree.
	Consider the quasi-projective variety Bir_d of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety Bir_do where the three polynomials have no common factor. In this talk we will compute their dimension and the decomposition in irreducible components. Then, we will show that Bir_d is connected for each d and that Bir_do is connected when d < 7.
Matthias LEUENBERGER Lie algebra generated by LNDs on surfaces given by {xy=p(z)}
	On n every volume preserving holomorphic automorphism can be approximated by compositions of flow maps of locally nilpotent derivations (LNDs). This follows by Andersén-Lempert theory from the fact that the Lie algebra generated by LNDs is equal to the Lie algebra of all volume preserving algebraic vector fields on n. We will see that this statement does not hold for the class of surfaces given by {xy=p(z)}. Moreover it is possible to give a precise description of the Lie algebra generated by LNDs.
Karol PALKA - Hunting for log-extremal curves on the example of Coolidge and Kumar-Murthy theorems

Marta PIEROPAN - Counting rational points on toric varieties via Cox rings

How to come

The journey to Enney is 2 hours from Geneva, 2h30 from Basel/Zurich, 1h30 from Lausanne.
See timetables on www.cff.ch.
The train station of Enney is at 10 minutes by foot from the center. See the map, with the blue path given by google (pay attention to the snow!)

Participants

Hamid Ahmadinhezhad (Linz)
Rafael Andrist (Wuppertal)
Bachar Al Hajjar (Dijon)
Ivan Arzhantsev (Moscow)
Ivan Bazhov (Moscow / Geneva)
Cinzia Bisi (Ferrara)
Jérémy Blanc (Basel)
Gavin Brown (Loughborough)
Alberto Calabri (Ferrara)
Jung Kyu Canci (Basel)
Fabrizio Donzelli (Ottawa)
Adrien Dubouloz (Dijon)
Jean-Philippe Furter (La Rochelle)
Polina Kotenkova (Moscow)
Isac Héden (Uppsala)
Seyed Reza Hosseini (Jena)
Anne-Sophie Kaloghiros (London)
Kevin Langlois (Grenoble)
Matthias Leuenberger (Bern)
Lucy Moser-Jauslin (Dijon)
Stefan Maubach (Bremen)
Karol Palka (Warsaw)
Alexander Perepechko (Grenoble/ Moscow)
Charlie Petitjean (Dijon)
Marta Pieropan (Munich)
Pierre-Marie Poloni (Basel)
Maria Fernanda Robayo (Basel)
Elena Romaskevich (Moscow)
Łukasz Sienkiewicz (Warsaw)
Immanuel Stampfli (Basel)
Ronan Terpereau (Grenoble)
Christian Urech (Basel)
Jaroslaw A. Wisniewski (Warsaw)
Susanna Zimmermann (Basel)

Organisers

Adrien Dubouloz (Dijon)
Jérémy Blanc (Basel)

Financial support

We gratefully acknowledge support from:
Swiss mathematical society
Swiss doctoral program
French ANR, project BIRPOL
University of Basel