some text Recent advances in Linear series
and Newton-Okounkov bodies
February 9-13, 2015 - Università degli Studi di Padova
Dipartimento di Matematica, Via Trieste 63, Room 1AD100 (first floor)

Workshop desctiption  
How to get here  
Tentative schedule  
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Ciro Ciliberto (Roma)
Alex Küronia (Budapest)
Victor Lozovanu (Milano)
Stefano Urbinati (Padova)

Local scientific committee:  

Bruno Chiarellotto
Ernesto Carlo Mistretta
Carla Novelli

Participants: Klaus Altmann (Berlin)
Francesco Bastianelli (Roma)
Cristiano Bocci (Siena)
Francesco Bottacin (Padova)
Alberto Calabri (Ferrara)
Sung Rak Choi (Pohang)
Pietro De Poi (Udine)
Thomas Dedieu (Toulouse)
Olivia Dumitrescu (Hannover)
Li Duo (Grenoble)
Lucja Farnik (Cracow)
Michal Farnik (Cracow)
Claudio Fontanari (Trento)
Jacopo Gandini (Pisa)
Brian Harbourne (Lincoln)
Giovanna Ilardi (Napoli)
Remke Kloosterman(Berlin)
Andreas Leopold Knutsen (Bergen/Rome)
John Lesieutre (Princeton)
Paolo Lella (Trento)
Luigi Lombardi (Bonn)
Catriona Maclean (Grenoble)
Carlo Martinucci (Padova)
Georg Merz (Göttingen)
Emilia Mezzetti (Trieste)
Rick Miranda (Fort Collins)
John Christian Ottem (Cambridge)
Jinhyung Park (Seoul)
Piotr Pokora (Cracow)
Elisa Postinghel (Leuven)
Joaquim Roé (Barcelona)
Xavier Roulleau (Poitiers)
David Schmitz (Marburg)
Constantin Shramov (Moscow)
Paola Supino (Roma)
Tomasz Szemberg (Cracow)
Justyna Szpond (Cracow)
Damiano Testa (Warwick)
Halszka Tutaj-Gasinska (Cracow)
Emanuele Ventura (Aalto)
Joonyeong Won (Daejeon)

Confirmed sponsors:

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----------------WORKSHOP DESCRIPTION----------------

Linear series are a basic and fundamental tool
in algebraic geometry. Many results and techniques
from their study are essential in different parts
of the field. There have been several new
developments in the last few years and it is almost
impossible to keep track of all of them.

A first example is given by the traditional Nagata
Conjecture, that can be expressed in asymptotic
terms using either Seshadri constants or Waldschmidt
constants. Recently it has become apparent that
Nagata Conjecture for points in the projective plane
is a manifestation of a much more general picture
concerning configurations of disjoint linear subspaces
of a projective space.

In a second instance, vector bundles have emerged
as powerful tools for analyzing linear series on curves
and surfaces. A classical result of this nature is
Reider's theorem that gives full conditions on surfaces
for adjoint line bundles to be globally generated. More
recently, the viewpoints and techniques of higher
dimensional geometry have started to play a central role.

One of the main focuses of the workshop will be on
Newton-Okounkov bodies. Their definition originates from
papers due to A. Okounkov in the 1990s. More recently,
Lazarsfeld and Mustaţă and independently
Kaveh and Khovanskii initiated an intensive research on
the topic recording strong relations of the construction
to properties of linear series that were not observed
at first.
The idea of the construction of Newton-Okounkov bodies
is to associate simple convex-geometric objects to linear
series on normal varieties. This idea comes as a
generalization of the toric case, where to each divisor
D one associates a polytope P_D. As for the toric, the
Newton-Okounkov body is a convex body which encodes
several properties of linear series, e.g. their volume.
In another direction, Dave Anderson showed that varieties
having an ample divisor with a rational polyhedral
Newton-Okounkov body with respect to some flag, admit a
flat degeneration to a toric variety, giving a nice way
of studying the initial variety from it's toric
degeneration. For linear series, Newton-Okounkov
bodies constitute a concise and accessible way to express
and control various asymptotic invariants, and measure
the positivity properties of the underlying divisors.

The aim of the workshop is to bring together experts in
the field and have a comparison of production in the
different possible directions of research.
The workshop will have a number of introductory talks in
the morning, whose goal is to stimulate further
discussions. In the afternoon there will be a division
into working groups on topics and issues chosen by the
participants themselves.

This methodology has already been tested in Warsaw in the
fall 2013 and gave excellent results both on a personal
level of the participants and at the mathematical level,
obtaining new collaborations and results during the same
week or later.

For more information please contact one of the organizers via e-mail.