7,659 research outputs found
Projective toric varieties as fine moduli spaces of quiver representations
This paper proves that every projective toric variety is the fine moduli
space for stable representations of an appropriate bound quiver. To accomplish
this, we study the quiver with relations corresponding to the
finite-dimensional algebra where
is a list of line bundles on a
projective toric variety . The quiver defines a smooth projective toric
variety, called the multilinear series , and a map . We provide necessary and sufficient conditions for the induced
map to be a closed embedding. As a consequence, we obtain a new geometric
quotient construction of projective toric varieties. Under slightly stronger
hypotheses on , the closed embedding identifies with the fine
moduli space of stable representations for the bound quiver .Comment: revised version: improved exposition, corrected typos and other minor
change
Cones of Hilbert functions
We study the closed convex hull of various collections of Hilbert functions.
Working over a standard graded polynomial ring with modules that are generated
in degree zero, we describe the supporting hyperplanes and extreme rays for the
cones generated by the Hilbert functions of all modules, all modules with
bounded a-invariant, and all modules with bounded Castelnuovo-Mumford
regularity. The first of these cones is infinite-dimensional and simplicial,
the second is finite-dimensional but neither simplicial nor polyhedral, and the
third is finite-dimensional and simplicial.Comment: 20 pages, 2 figure
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
Syzygies, multigraded regularity and toric varieties
Using multigraded Castelnuovo-Mumford regularity, we study the equations
defining a projective embedding of a variety X. Given globally generated line
bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle
L := B_1^m_1 \otimes ... \otimes B_k^m_k. We give conditions on the m_i which
guarantee that the ideal of X in P(H^0(X,L)) is generated by quadrics and the
first p syzygies are linear. This yields new results on the syzygies of toric
varieties and the normality of polytopes.Comment: improved exposition and corrected typo
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