3,658 research outputs found
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
A comment on free-fermion conditions for lattice models in two and more dimensions
We analyze free-fermion conditions on vertex models. We show --by examining
examples of vertex models on square, triangular, and cubic lattices-- how they
amount to degeneration conditions for known symmetries of the Boltzmann
weights, and propose a general scheme for such a process in two and more
dimensions.Comment: 12 pages, plain Late
Standard monomial theory for wonderful varieties
A general setting for a standard monomial theory on a multiset is introduced
and applied to the Cox ring of a wonderful variety. This gives a degeneration
result of the Cox ring to a multicone over a partial flag variety. Further, we
deduce that the Cox ring has rational singularities.Comment: v3: 20 pages, final version to appear on Algebras and Representation
Theory. The final publication is available at Springer via
http://dx.doi.org/10.1007/s10468-015-9586-z. v2: 20 pages, examples added in
Section 3 and in Section
Symplectomorphism group relations and degenerations of Landau-Ginzburg models
In this paper, we describe explicit relations in the symplectomorphism groups
of toric hypersurfaces. To define the elements involved, we construct a proper
stack of toric hypersurfaces with compactifying boundary representing toric
hypersurface degenerations. Our relations arise through the study of the one
dimensional strata of this stack. The results are then examined from the
perspective of homological mirror symmetry where we view sequences of relations
as maximal degenerations of Landau-Ginzburg models. We then study the B-model
mirror to these degenerations, which gives a new mirror symmetry approach to
the minimal model program.Comment: 100 pages, 24 figure
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