Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

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Existence of minimal models for varieties of log general type

Parameter spaces: constructions and examples * Basic facts about moduli spaces of curves * Techniques * Construction of M_g * Limit Linear Series and the Brill-Noether Theory * Geometry of moduli spaces: Selected Results.

Moduli of curves

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Compactifying the space of stable maps

Given a smooth complex Deligne-Mumford stack ${\cal X}$ with a projective coarse moduli space, we introduce Gromov-Witten invariants of ${\cal X}$ and prove some of their basic properties, including the WDVV equation.

Gromov-Witten theory of Deligne-Mumford stacks

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

We study the structure of the stacks of twisted stable maps to the classifying stack of a finite group G—which we call the stack of twisted G-covers, or twisted G-bundles. For a suitable group Gwe show that the substack corresponding to admissible G-covers is a smooth projective fine moduli space. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Twisted Bundles and Admissible Covers

Nous introduisons et etudions, en caracteristique positive et mixte, une classe de champs algebriques a champs d'inertie finis, appeles champs algebriques moderes. Cette classe inclue les champs moderes de Deligne-Mumford, et on peut dire qu'elle se comporte mieux que la classe des champs generaux de Deligne-Mumford. En outre, nous donnons une caracterisation complete des schemas en groupes finis plats et lineairement reductifs sur une base quelconque. Notre resultat principal est le suivant : un champ algebrique modere est, localement dans la topologie etale, un quotient par une action d'un schema en groupes fini lineairement reductif.

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Tame stacks in positive characteristic

Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Such a factorization exists which is functorial with respect to absolute isomorphisms, and compatible with a normal crossings divisor. The same holds for algebraic and analytic spaces. Another proof of the main theorem by the fourth author appeared in math.AG/9904076.

Dan Abramovich

Papers

Compactifying the space of stable maps

Gromov-Witten theory of Deligne-Mumford stacks

Twisted Bundles and Admissible Covers

Tame stacks in positive characteristic

Torification and Factorization of Birational Maps