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58
Tropical curves, their Jacobians and theta functions
, 2006
"... We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the AbelJacobi, RiemannRoch and Riemann theta divisor theorems. ..."
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Cited by 29 (3 self)
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We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the AbelJacobi, RiemannRoch and Riemann theta divisor theorems.
Affine Manifolds, Log Structures, and Mirror Symmetry
, 2003
"... We outline work in progress suggesting an algebrogeometric version of the StromingerYauZaslow conjecture. We define the notion of a toric degeneration, a special case of a maximally unipotent degeneration of CalabiYau manifolds. We then show how in this case the dual intersection complex has a n ..."
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Cited by 22 (4 self)
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We outline work in progress suggesting an algebrogeometric version of the StromingerYauZaslow conjecture. We define the notion of a toric degeneration, a special case of a maximally unipotent degeneration of CalabiYau manifolds. We then show how in this case the dual intersection complex has a natural structure of an affine manifold with singularities. If the degeneration is polarized, we also obtain an intersection complex, also an affine manifold with singularities, related by a discrete Legendre transform to the dual intersection complex. Finally, we introduce log structures as a way of reversing this construction: given an affine manifold with singularities with a suitable polyhedral decomposition, we can produce a degenerate CalabiYau variety along with a log structure. Hopefully, in interesting cases, this object will have a wellbehaved deformation theory, allowing us to use the discrete Legendre transform to construct mirror pairs of CalabiYau manifolds. We also connect this approach to the topological form of the StromingerYauZaslow conjecture.
Toric Hilbert schemes
 Duke Math. J
, 1999
"... Abstract: We introduce and study the toric Hilbert scheme which parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal. 1. ..."
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Cited by 21 (3 self)
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Abstract: We introduce and study the toric Hilbert scheme which parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal. 1.
A point set whose space of triangulations is disconnected
 J. AMER. MATH. SOC
, 2000
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Compactification of the moduli space of hyperplane arrangements
 J. Algebraic Geom
"... Consider the moduli space M 0 of arrangements of n hyperplanes in general position in projective (r − 1)space. When r = 2 the space has a compactification given by the moduli space of stable curves of genus 0 with n marked points. In higher dimensions, the analogue of the moduli space of stable cur ..."
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Cited by 19 (6 self)
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Consider the moduli space M 0 of arrangements of n hyperplanes in general position in projective (r − 1)space. When r = 2 the space has a compactification given by the moduli space of stable curves of genus 0 with n marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs (S, B) consisting of a variety S (possibly reducible) and a divisor B = B1 +.. + Bn, satisfying various additional conditions. We identify the closure of M 0 in the moduli space of stable pairs as Kapranov’s Hilbert quotient compactification of M 0, and give an explicit description of the pairs at the boundary. We also construct additional irreducible components of the moduli space of stable pairs.
From real affine geometry to complex geometry
"... Abstract. We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of CalabiYau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical orderbyorder descrip ..."
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Cited by 17 (3 self)
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Abstract. We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of CalabiYau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical orderbyorder description of the degeneration via families of tropical trees. This gives complete control of the Bmodel side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. This
Nonconnected toric Hilbert schemes
, 2001
"... Abstract. We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc. 13:3 (2000), 611–637 not only in size, but also in that the associated toric Hilbert schemes are no ..."
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Cited by 13 (3 self)
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Abstract. We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc. 13:3 (2000), 611–637 not only in size, but also in that the associated toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5dimensional polytopes with nonconnected graph of triangulations.
Computational approaches to lattice packing and covering problems
 Discrete Comput. Geom. 35 (2006) 73–116. MR2183491 (2006k:52048
"... We describe algorithms which solve two classical problems in lattice geometry for any fixed dimension d: the lattice covering and the simultaneous lattice packing–covering problem. Both algorithms involve semidefinite programming and are based on Voronoi’s reduction theory for positive definite quad ..."
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Cited by 12 (6 self)
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We describe algorithms which solve two classical problems in lattice geometry for any fixed dimension d: the lattice covering and the simultaneous lattice packing–covering problem. Both algorithms involve semidefinite programming and are based on Voronoi’s reduction theory for positive definite quadratic forms which describes all possible Delone triangulations of Z d. Our implementations verify all known results in dimensions d ≤ 5. Beyond that we attain complete lists of all locally optimal solutions for d = 5. For d = 6 our computations produce new best known covering as well as packing–covering
Stable pair, tropical, and log canonical compact moduli of del Pezzo surfaces
"... Abstract. We give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the GrothendieckKnudsen compactification M0,n ⊂ M0,n. §1. Introduction and statement of results Throughout we work over an algebraically closed field k. Let Y n be the ..."
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Cited by 10 (4 self)
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Abstract. We give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the GrothendieckKnudsen compactification M0,n ⊂ M0,n. §1. Introduction and statement of results Throughout we work over an algebraically closed field k. Let Y n be the moduli space of smooth marked del Pezzo surfaces S of degree 9−n. We begin by observing that each such surface comes with a natural boundary: The marking of S induces a labelling of its (−1)curves, and we can take their union B ⊂ S. This gives
Stable reductive varieties, I: Affine varieties
 Invent. Math
"... 1. Main definitions and results 3 2. General criteria 6 2.1. Seminormality and connectedness of isotropy groups 6 ..."
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Cited by 9 (7 self)
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1. Main definitions and results 3 2. General criteria 6 2.1. Seminormality and connectedness of isotropy groups 6