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This book provides an overview of the latest progress on
rationality questions in algebraic geometry. It discusses new
developments such as universal triviality of the Chow group of zero
cycles, various aspects of stable birationality, cubic and Fano
fourfolds, rationality of moduli spaces and birational invariants
of group actions on varieties, contributed by the foremost experts
in their fields. The question of whether an algebraic variety can
be parametrized by rational functions of as many variables as its
dimension has a long history and played an important role in the
history of algebraic geometry. Recent developments in algebraic
geometry have made this question again a focal point of research
and formed the impetus to organize a conference in the series of
conferences on the island of Schiermonnikoog. The book follows in
the tradition of earlier volumes, which originated from conferences
on the islands Texel and Schiermonnikoog.
This book provides an overview of the latest developments
concerning the moduli of K3 surfaces. It is aimed at algebraic
geometers, but is also of interest to number theorists and
theoretical physicists, and continues the tradition of related
volumes like "The Moduli Space of Curves" and "Moduli of Abelian
Varieties," which originated from conferences on the islands Texel
and Schiermonnikoog and which have become classics. K3 surfaces and
their moduli form a central topic in algebraic geometry and
arithmetic geometry, and have recently attracted a lot of attention
from both mathematicians and theoretical physicists. Advances in
this field often result from mixing sophisticated techniques from
algebraic geometry, lattice theory, number theory, and dynamical
systems. The topic has received significant impetus due to recent
breakthroughs on the Tate conjecture, the study of stability
conditions and derived categories, and links with mirror symmetry
and string theory. At the same time, the theory of irreducible
holomorphic symplectic varieties, the higher dimensional analogues
of K3 surfaces, has become a mainstream topic in algebraic
geometry. Contributors: S. Boissiere, A. Cattaneo, I. Dolgachev, V.
Gritsenko, B. Hassett, G. Heckman, K. Hulek, S. Katz, A. Klemm, S.
Kondo, C. Liedtke, D. Matsushita, M. Nieper-Wisskirchen, G.
Oberdieck, K. Oguiso, R. Pandharipande, S. Rieken, A. Sarti, I.
Shimada, R. P. Thomas, Y. Tschinkel, A. Verra, C. Voisin.
This book provides an overview of the latest progress on
rationality questions in algebraic geometry. It discusses new
developments such as universal triviality of the Chow group of zero
cycles, various aspects of stable birationality, cubic and Fano
fourfolds, rationality of moduli spaces and birational invariants
of group actions on varieties, contributed by the foremost experts
in their fields. The question of whether an algebraic variety can
be parametrized by rational functions of as many variables as its
dimension has a long history and played an important role in the
history of algebraic geometry. Recent developments in algebraic
geometry have made this question again a focal point of research
and formed the impetus to organize a conference in the series of
conferences on the island of Schiermonnikoog. The book follows in
the tradition of earlier volumes, which originated from conferences
on the islands Texel and Schiermonnikoog.
This book provides an overview of the latest developments
concerning the moduli of K3 surfaces. It is aimed at algebraic
geometers, but is also of interest to number theorists and
theoretical physicists, and continues the tradition of related
volumes like "The Moduli Space of Curves" and "Moduli of Abelian
Varieties," which originated from conferences on the islands Texel
and Schiermonnikoog and which have become classics. K3 surfaces and
their moduli form a central topic in algebraic geometry and
arithmetic geometry, and have recently attracted a lot of attention
from both mathematicians and theoretical physicists. Advances in
this field often result from mixing sophisticated techniques from
algebraic geometry, lattice theory, number theory, and dynamical
systems. The topic has received significant impetus due to recent
breakthroughs on the Tate conjecture, the study of stability
conditions and derived categories, and links with mirror symmetry
and string theory. At the same time, the theory of irreducible
holomorphic symplectic varieties, the higher dimensional analogues
of K3 surfaces, has become a mainstream topic in algebraic
geometry. Contributors: S. Boissiere, A. Cattaneo, I. Dolgachev, V.
Gritsenko, B. Hassett, G. Heckman, K. Hulek, S. Katz, A. Klemm, S.
Kondo, C. Liedtke, D. Matsushita, M. Nieper-Wisskirchen, G.
Oberdieck, K. Oguiso, R. Pandharipande, S. Rieken, A. Sarti, I.
Shimada, R. P. Thomas, Y. Tschinkel, A. Verra, C. Voisin.
The Handbook of Moduli, comprising three volumes, offers a
multi-faceted survey of a rapidly developing subject aimed not just
at specialists but at a broad community of producers of algebraic
geometry, and even at some consumers from cognate areas. The
thirty-five articles in the Handbook, written by fifty leading
experts, cover nearly the entire range of the field. They reveal
the relations between these many threads and explore their
connections to other areas of algebraic geometry, number theory,
differential geometry, and topology. The goals of the Handbook are
to introduce the techniques, examples, and results essential to
each topic, and to say enough about recent developments to provide
a gateway to the primary sources. Many articles are original
treatments commissioned to bridge gaps in the literature and to
make important problems accessible to a wide audience for the first
time, and many others illustrate yogas and heuristics that experts
use privately to guide intuition or simplify calculation, but that
do not appear in published work aimed at other specialists. This is
the first of three volumes constituting the Handbook of Moduli, and
is also available as part of a three volume set.
The Handbook of Moduli, comprising three volumes, offers a
multi-faceted survey of a rapidly developing subject aimed not just
at specialists but at a broad community of producers of algebraic
geometry, and even at some consumers from cognate areas. The
thirty-five articles in the Handbook, written by fifty leading
experts, cover nearly the entire range of the field. They reveal
the relations between these many threads and explore their
connections to other areas of algebraic geometry, number theory,
differential geometry, and topology. The goals of the Handbook are
to introduce the techniques, examples, and results essential to
each topic, and to say enough about recent developments to provide
a gateway to the primary sources. Many articles are original
treatments commissioned to bridge gaps in the literature and to
make important problems accessible to a wide audience for the first
time, and many others illustrate yogas and heuristics that experts
use privately to guide intuition or simplify calculation, but that
do not appear in published work aimed at other specialists. This is
a set comprising the following volumes: Handbook of Moduli: Volume
I (vol. 24 of the ALM series) Handbook of Moduli: Volume II (vol.
25 of the ALM series) Handbook of Moduli: Volume III (vol. 26 of
the ALM series)
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