Fano varieties and their automorphisms

Start: Tuesday 30th October 2018, at 13:00.  
End: Friday: 2nd November 2018, at 13:00.
All talks, except the colloquium, will take place in SCH0.13 (Schofield Building room 0.13). 

Speakers - Titles:
  • Jeremy Blanc (Basel) - Abelian quotients of the Cremona groups in higher dimension
  • Alberto Calabri (Ferrara) - On the factorisation of plane Cremona maps
  • Ivan Cheltsov (Edinburgh) - On a question of Charlie Stibitz and Ziquan Zhuang
  • Enrico Fatighenti (Loughborough) - Fano varieties of K3 type and IHS manifolds
  • Isac Hedén (Warwick) - Borel- and maximal subgroups of the plane Cremona group
  • Karol Palka (Warsaw) - The Generalized Jacobian Conjecture
  • Jean-Philippe Furter (Bordeaux) - Lengths in the Cremona group
  • Anne-Sophie Kaloghiros (Brunel) - Threefold Calabi-Yau pairs
  • Hanspeter Kraft (Basel) - The regularization theorem of A. Weil
  • Anne Lonjou (Basel) - Cremona group and hyperbolic spaces
  • Miles Reid (Warwick) - Some explicit deformations
  • Christian Urech (Imperial College London) - On the characterization of affine surfaces by their automorphism groups
  • Susanna Zimmermann (Angers) - Abelian quotients of the Cremona groups in higher dimension

Schedule:  
          
 Tuesday        
 WednesdayThursday    Friday
 10:00 - 11:00
 Zimmermann (II)
 10:00 - 11:00
Kraft
9:00 - 10:00 
Calabri
 
coffee break (SCH0.04)coffee break (SCH0.04)
13:00 - 14:00
Furter 
11:10 - 12:10
Reid
11:30 - 12:30
Blanc (I) 
10:30 - 11:30
Kaloghiros 
 coffee break
(SCH0.04)
IAS lunch 
Council Chamber, Hazlerigg Building
lunch 
(SCH0.04)
11:45 - 12:45
Fatighenti 
14:30 - 15:30
Hedén 
14:00 - 15:00 
Cheltsov
 14:00 - 15:00
Urech
coffee break (SCH0.04) 
 lunch 
(SCH0.04)
16:00 - 17:00
Zimmermann (I)
 15:00 - 16:00
Palka
 15:30 - 16:30
Lonjou
 
17:30 - 18:30 (Room SCH001)
Hanspeter Kraft
Departmental Colloquium Talk
followed by reception (SCH0.04)
 free afternoon
suggestion: Diwali
17:00 - 18:00 
Blanc (II) 

Dinner at 6:30
 


Registration: Please send an email to H.Ahmadinezhad@lboro.ac.uk

This event is a part of the activities for the semester in Geometry sponsored by the Loughborough University Institute of Advanced Studies.

Abstracts:

Blanc-Zimmermann
Title: Abelian quotients of the Cremona groups in higher dimension
Abstract: We study groups of birational selfmaps $\Bir(X)$, where $X$ is a rationally connected variety of dimension at least $3$, over an algebraically closed field of characteristic zero. We are interested in the situation where these groups are large. 
One prominent case is when $X$ is the projective space $\mathbb{P}^n$, in which case $\Bir(X)$ is the Cremona group of rank $n$. We produce infinitely many distinct group homomorphisms from $\Bir(\p^n)$ to $\Z/2$, showing that the Cremona group is not perfect hence not simple. As a consequence we also obtain that the Cremona group of rank $3$ is not generated by linear and Jonqui\`eres elements.

Calabri
Title: On the factorisation of plane Cremona maps
Abstract: Let n(f) be the minimum number of standard quadratic maps needed for factorising a plane Cremona map f.
I will report on a joint work in progress with Jérémy Blanc and Julie Déserti about the computation of n(f), in particular about the upper bound for n(f) when f is a plane Cremona map of fixed degree d.

Cheltsov
Title: On a question of Charlie Stibitz and Ziquan Zhuang
Abstract:  In https://arxiv.org/abs/1802.08381, Charlie Stibitz and Ziquan Zhuang posed a nice question: 
is it true that alpha-invariants of birationally superrigid Fano varieties are greater than 1/2? 
In my talk, I will explain that the answer to this question is probably No. 
This is a joint work with Ziquan Zhuang (Princeton).

Fatighenti
Title: Fano varieties of K3 type and IHS manifolds
Abstract: Subvarieties of Grassmannian (and especially Fano varieties) obtained from section of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type, for their deep links with hyperk\"ahler geometry. This talk will be mainly devoted to the construction of some new examples of such varieties. This is a work in progress with Giovanni Mongardi.

Hedén
Title: Borel- and maximal subgroups of the plane Cremona group
Abstract: All Borel subgroups of a linear algebraic group are conjugate. I will briefly address the question whether this property holds also for Cremona groups (quoting some results of V. Popov), and then talk about the group of plane Cremona transformations whose Jacobian determinant is a cube. This interesting group turns out to be very large. This is a joint work in progress with J.-P. Furter.

Palka
Title: The Generalized Jacobian Conjecture
Abstract: A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. Since the conjecture for the affine plane is open, it is worth studying what is going on for surfaces similar to it from the topological and algebraic point of view. I will discuss recent progress in this direction (obtained with Adrien Dubouloz), including some surprising counterexamples. https://arxiv.org/abs/1701.01425

Furter
Title: Lengths in the Cremona group
Abstract: Define the length of a plane Cremona transformation $f$ as the least
nonnegative integer $k$ for which there exists a decomposition
$f = a_1 \circ b_1 \circ  \cdots \circ a_k \circ b_k \circ a_{k+1}$
where the elements $a_i$ are automorphisms of the projective plane and the elements $b_i$
are Jonqui\`eres transformations. We explain how to compute this length and give,
among others, some applications to the fields of: dynamics, monomial transformations, and
filtrations of the Cremona group. This is a joint work with Jérémy Blanc.

Kaloghiros
Title: Threefold Calabi-Yau pairs
Abstract: The dimension of the dual complex of a Calabi-Yau pair is a crepant birational invariant. A CY pair for which this invariant is maximal is said to have maximal intersection. Pairs of maximal intersection are degenerate objects, they have Fano-type behaviour in a suitable sense. In dimension 2, these are precisely the CY pairs that have a toric model. 
I will discuss maximal intersection CY pairs in dimension 3 and present some examples.

Kraft 
Title: The regularization theorem of A. Weil
Abstract: We give a modern proof of the Regularization Theorem of Andr\'e Weil which says that for every
rational action of an algebraic group $G$ on a variety $X$ there exist a variety $Y$ with a regular action of $G$ and 
a $G$-equivariant birational map $X \dto Y$. Moreover, we show that a rational action of $G$ on an affine variety $X$ with the property that each $g$ from a dense subgroup of $G$ induces a regular automorphism of $X$, is a regular action.

Lonjou
Title: Cremona group and hyperbolic spaces
Abstract: The Cremona group is the group of birational transformations of the projective plane. It acts on a hyperbolic space which is an infinite dimensional version of the hyperboloid model of H^n. This action is the main recent tool to study the Cremona group. After defining it, we will study its Voronoï tesselation, and describe some graphs naturally associated with this construction. Finally we will discuss which of these graphs are Gromov-hyperbolic. 

Reid
Title: Some explicit deformations
Abstract: I outline a graded ring approach to constructing QQ-Gorenstein deformation families related to "singularity content" and "mutation" of polygons. For example, the weighted projective planes PP(a^2,b^2,c^2) for (a,b,c) a Markov trip are known to smooth to PP^2 by work of Manetti and Hacking. Nowadays, this interpreted in terms of mirror symmetry, but my point of view is confined to the B side. Based on experimental cases, I believe that the graded ring corresponding to the (abc)th Veronese truncation v_abc(PP(a^2,b^2,c^2)) (its 1/3-anticanonical ring) has a k-dimensional unobstructed smoothing that successively cancels the mutations of the Markov spectrum. The calculations are similar in feel to the easier cases of diptych varieties.

Urech
Title: On the characterization of affine surfaces by their automorphism groups
Abstract: In this talk we will look at the question, in as far affine surfaces are characterized by the group structure of their automorphism groups. In particular, I will show that if $S$ is a toric surface and $S'$ any normal affine surface such that $Aut(S)$ and $Aut(S')$ are isomorphic as groups, then $S$ and $S'$ are isomorphic. The proof involves birational techniques as well as results from toric geometry. This is joint work with Liendo and Regeta.