This volume has its origins in an unusual conference held in July, 2004, in
Snowbird, Utah. The conference was an experiment, designed to address a well-
recognized phenomenon in algebraic geometry. Namely, our subject has become
broad and deep enough that most graduate students limit their training to a narrow
area of it in order to produce original research in a reasonable time. Thus the early
postdoctoral years are an ideal time to broaden knowledge and build connections,
both personal and intellectual, that will nourish a lifelong research career. The
conference was intended to give that opportunity to recent Ph.D.'s, who had already
developed an area of expertise and were now ready to broaden their horizons. Our
goal was to expose those who represent the future of our subject to the ideas and
problems central to parts of the field other than their own. In addition, there was
ample time for small group discussions and networking, to encourage these young
researchers to meet future colleagues and collaborators.
We selected ten of the participants to give lectures in the mornings. The goal of
the lectures was to introduce everyone to a wide swathe of algebraic geometry, and
to give them a working familiarity with exciting parts of the field far removed from
their thesis work. In the afternoons, the participants broke up into eight working
groups, each led by a more senior mentor, to study in more depth (and in a hands-
on fashion) a new part of the subject. The conference was a resounding success, and
not through any effort of the organizers: by bringing together top young talent in
the field, it was inevitable that the discussions between lectures and in the evenings
should lead to exciting mathematics. The eight mentors also served as catalysts,
and we were lucky to have them: Linda Chen, Gabi Farkas, Angela Gibney, Allen
Knutson, Sandor Kovacs, Diane Maclagan, Mike Nakamaye, and Tony Pantev. We
strongly hope this nontraditional experiment is repeated periodically in the future.
We initially had no intention of having a proceedings volume, since we felt that
the subject did not need yet another publication for the sake of publication. But
the quality of the lectures was so uniformly high that it was obviously a shame that
other people could not benefit for them. We informally polled the audience and the
mentors, and there was overwhelming popular demand for the speakers to record
their thoughts for posterity.
Not all the articles in this volume arose from lectures. The article by Kovacs
et al. is based on the efforts of the working group in higher-dimensional geometry,
and may give the reader some idea of the flavor of the afternoon sessions. Also,
Caldararu's article relates to the topic he discussed at Snowbird, but is based on
the lecture series he gave later that year at the autumn school in Lukecin, Poland.
I would like to warmly thank my co-organizers, Herb Clemens and Rob Lazars-
feld, for their advice and assistance, in both the planning and execution of the
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