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Complete moduli of cubic threefolds and their intermediate Jacobians
The intermediate Jacobian map, which associates to a smooth cubic threefold
its intermediate Jacobian, does not extend to the GIT compactification of the
space of cubic threefolds, not even as a map to the Satake compactification of
the moduli space of principally polarized abelian fivefolds. A much better
"wonderful" compactification of the space of cubic threefolds was constructed
by the first and fourth authors --- it has a modular interpretation, and
divisorial normal crossing boundary. We prove that the intermediate Jacobian
map extends to a morphism from the wonderful compactification to the second
Voronoi toroidal compactification of the moduli of principally polarized
abelian fivefolds --- the first and fourth author previously showed that it
extends to the Satake compactification. Since the second Voronoi
compactification has a modular interpretation, our extended intermediate
Jacobian map encodes all of the geometric information about the degenerations
of intermediate Jacobians, and allows for the study of the geometry of cubic
threefolds via degeneration techniques. As one application we give a complete
classification of all degenerations of intermediate Jacobians of cubic
threefolds of torus rank 1 and 2.Comment: 56 pages; v2: multiple updates and clarification in response to
detailed referee's comment
Hurwitz numbers and intersections on moduli spaces of curves
This article is an extended version of preprint math.AG/9902104. We find an
explicit formula for the number of topologically different ramified coverings
of a sphere by a genus g surface with only one complicated branching point in
terms of Hodge integrals over the moduli space of genus g curves with marked
points.Comment: 30 pages (AMSTeX). Minor typos are correcte
Enumerative aspects of the Gross-Siebert program
We present enumerative aspects of the Gross-Siebert program in this
introductory survey. After sketching the program's main themes and goals, we
review the basic definitions and results of logarithmic and tropical geometry.
We give examples and a proof for counting algebraic curves via tropical curves.
To illustrate an application of tropical geometry and the Gross-Siebert program
to mirror symmetry, we discuss the mirror symmetry of the projective plane.Comment: A version of these notes will appear as a chapter in an upcoming
Fields Institute volume. 81 page
Compact moduli of plane curves
We construct a compactification M_d of the moduli space of plane curves of
degree d. We regard a plane curve C as a surface-divisor pair (P^2,C) and
define M_d as a moduli space of pairs (X,D) where X is a degeneration of the
plane. We show that, if d is not divisible by 3, the stack M_d is smooth and
the degenerate surfaces X can be described explicitly.Comment: 46 pages. Final version to be published in Duke Mathematical Journa
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