68 research outputs found
Polyhedral structures on tropical varieties
Tropical varieties are polyhedral shadows of classical varieties. The purpose
of these expository notes is to explain the origin of this polyhedral complex
structure from the perspective of Gr\"obner bases. To appear in the proceedings
of the 2011 Bellairs Workshop in Number Theory
Lower and upper bounds for nef cones
The nef cone of a projective variety Y is an important and often elusive
invariant. In this paper we construct two polyhedral lower bounds and one
polyhedral upper bound for the nef cone of Y using an embedding of Y into a
toric variety. The lower bounds generalize the combinatorial description of the
nef cone of a Mori dream space, while the upper bound generalizes the
F-conjecture for the nef cone of the moduli space \bar{M}_{0,n} to a wide class
of varieties.Comment: 25 pages, 4 figures. Final version to appear in IMR
Uniform bounds on multigraded regularity
We give an effective uniform bound on the multigraded regularity of a
subscheme of a smooth projective toric variety X with a given multigraded
Hilbert polynomial. To establish this bound, we introduce a new combinatorial
tool, called a Stanley filtration, for studying monomial ideals in the
homogeneous coordinate ring of X. As a special case, we obtain a new proof of
Gotzmann's regularity theorem. We also discuss applications of this bound to
the construction of multigraded Hilbert schemes.Comment: 23 pages, 2 figure
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