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Current Developments in Algebraic Geometry by Lucia Caporaso, James McKernan, Mircea Mustata

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About this book :-
Current Developments in Algebraic Geometry written by Lucia Caporaso, James McKernan, Mircea Mustata
Algebraic geometry is one of the most diverse fields of research in mathematics. It has had an incredible evolution over the past century, with new subfields constantly branching off and spectacular progress in certain directions, and at the same time, with many fundamental unsolved problems still to be tackled. In the spring of 2009 the first main workshop of the MSRI algebraic geometry program served as an introductory panorama of current progress in the field, addressed to both beginners and experts. This volume reflects that spirit, offering expository overviews of the state of the art in many areas of algebraic geometry. Prerequisites are kept to a minimum, making the book accessible to a broad range of mathematicians. Many chapters present approaches to long-standing open problems by means of modern techniques currently under development and contain questions and conjectures to help spur future research.
The articles gathered in this volume are contributed from many of Joe’s colleagues, postdoctoral mentees and students. The topics reflect the breadth of Joe’s own interests, indeed the breadth of “classical” algebraic geometry in our postclassical era. From all of us whom Joe has led and inspired, we dedicate this volume to Joe as a token of our gratitude.
The conference was funded by the National Science Foundation (DMS-1045217), the Clay Mathematics Institute, and Harvard University. Its success is due in large part to the tireless efforts of Irene Minder and the rest of the staff of the Harvard Mathematics Department. We are grateful to Izzet Coskun for his contributions to the planning of the conference, as well as the other members of the organizing committee (David Ellwood, Benedict Gross, and David Smyth) and the scientific committee (Dan Abramovich, Lucia Caporaso, Kieran O’Grady, and Rahul Pandharipande).

Book Detail :-
Title: Current Developments in Algebraic Geometry
Edition:
Author(s): Lucia Caporaso, James McKernan, Mircea Mustata
Publisher: Cambridge University Press
Series: Mathematical Sciences Research Institute Publications
Year: 2012
Pages: 613
Type: PDF
Language: English
ISBN: 052176825X,9780521768252
Country: US
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About Author :-
The author James McKernan (born 1964) is a mathematician, and a professor of mathematics at the University of California, San Diego. He was a professor at MIT from 2007 until 2013.
McKernan was educated at the Campion School, Hornchurch, and Trinity College, Cambridge, before going on to earn his Ph.D. from Harvard University in 1991. His dissertation, On the Hyperplane Sections of a Variety in Projective Space, was supervised by Joe Harris.
McKernan was the joint winner of the Cole Prize in 2009, and joint recipient of the Clay Research Award in 2007. Both honors were received jointly with his colleague Christopher Hacon. He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Algebraic Geometry". He was the joint winner (with Christopher Hacon) of the 2018 Breakthrough Prize in Mathematics.
He was elected as a Fellow of the American Mathematical Society in the 2020 Class, for "contributions to algebraic geometry, in particular his proof of the finite generation of the canonical ring, the existence of flips and the boundedness of varieties of log general type".[7

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Book Contents :-
Current Developments in Algebraic Geometry written by Lucia Caporaso, James McKernan, Mircea Mustata cover the following topics.
Introduction
Classification of 2-Fano manifolds with high index by Carolina Araujo and Ana-Maria Castravet
Abelian varieties associated to Gaussian lattices by Arnaud Beauville
Rank of divisors on graphs: An algebro-geometric analysis by Lucia Caporaso
A tour of stable reduction with applications by Sebastian Casalaina-Martin
The solvable monodromy extension property and varieties of log general type by Sabin Cautis
Effective divisors of moduli spaces of curves and abelian varieties by Dawei Chen, Gavril Farkas, and Ian Morrison
Rational self maps of Calabi-Yau manifolds by Xi Chen
Variations on Nagata’s conjecture Ciro Ciliberto, Brian Harbourne, Rick Miranda, and Joaquim by Ro´e
Symplectic restriction varieties and geometric branching rules by Izzet Coskun
Riemann-Roch for Deligne-Mumford stacks by Dan Edidin
The regularity of the conductor by David Eisenbud and Bernd Ulrich
Stability of genus five canonical curves by Maksym Fedorchuk and David Ishii Smyth
Restriction of sections for families of abelian varieties by Tom Graber and Jason Michael Starr
Correspondence and cycles spaces: A result comparing their cohomologies by Mark Green and Phillip Griffiths
Geometry of theta divisors—a survey by Samuel Grushevsky and Klaus Hulek
Singular curves and their compactified Jacobians by Jesse Leo Kass
On the G¨ottsche threshold Steven L. Kleiman and Vivek V. Shende, with an appendix by Ilya Tyomkin
Curve counting `a la G¨ottsche by Steven L. Kleiman
Mn¨ev-Sturmfels universality for schemes by Seok Hyeong Lee and Ravi Vakil
Gromov-Witten theory and Noether-Lefschetz theory by Davesh Maulik and Rahul Pandharipande
Numerical Macaulification by Juan Migliore and Uwe Nagel
The non-nef locus in positive characteristic by Mircea Mustat¸?a
Pairwise incident planes and hyperk¨ahler four-folds by Kieran G. O’Grady
Derived equivalence and non-vanishing loci by Mihnea Popa
Degenerations of rationally connected varieties and PAC fields by Jason Starr
Remarks on curve classes on rationally connected varieties by Claire Voisin


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