3,408 research outputs found
Mutations of Laurent Polynomials and Flat Families with Toric Fibers
We give a general criterion for two toric varieties to appear as fibers in a
flat family over the projective line. We apply this to show that certain
birational transformations mapping a Laurent polynomial to another Laurent
polynomial correspond to deformations between the associated toric varieties
Polarized Complexity-One T-Varieties
We describe polarized complexity-one T-varieties combinatorially in terms of
so-called divisorial polytopes, and show how geometric properties of such a
variety can be read off the corresponding divisorial polytope. We compare our
description with other possible descriptions of polarized complexity-one
T-varieties. We also describe how to explicitly find generators of affine
complexity-one T-varieties.Comment: 18 pages, 3 figure
Vanishing cotangent cohomology for Pl\"ucker algebras
We use representation theory and Bott's theorem to show vanishing of higher
cotangent cohomology modules for the homogeneous coordinate ring of
Grassmannians in the Pl\"ucker embedding. As a biproduct we answer a question
of Wahl about the cohomology of the square of the ideal sheaf for the case of
Pl\"ucker relations.Comment: Some results generalized to isotropic Grassmannian
Fano Schemes for Generic Sums of Products of Linear Forms
We study the Fano scheme of -planes contained in the hypersurface cut out
by a generic sum of products of linear forms. In particular, we show that under
certain hypotheses, linear subspaces of sufficiently high dimension must be
contained in a coordinate hyperplane. We use our results on these Fano schemes
to obtain a lower bound for the product rank of a linear form. This provides a
new lower bound for the product ranks of the Pfaffian and permanent, as well as giving a new proof that the product and tensor ranks
of the determinant equal five. Based on our results, we formulate
several conjectures.Comment: 22 pages. v2: minor revisions to v
Deformations of Rational T-Varieties
We show how to construct certain homogeneous deformations for rational normal
varieties with codimension one torus action. This can then be used to construct
homogeneous deformations of any toric variety in arbitrary degree. For locally
trivial deformations coming from this construction, we calculate the image of
the Kodaira-Spencer map. We then show that for a smooth complete toric variety,
our homogeneous deformations span the space of first-order deformations.Comment: 22 pages, 9 figures; v2 minor changes to introduction; v3 some
corrections, to appear in Journal of Algebraic Geometr
The Geometry of T-Varieties
This is a survey of the language of polyhedral divisors describing
T-varieties. This language is explained in parallel to the well established
theory of toric varieties. In addition to basic constructions, subjects touched
on include singularities, separatedness and properness, divisors and
intersection theory, cohomology, Cox rings, polarizations, and equivariant
deformations, among others.Comment: 42 pages, 17 figures. v2: minor changes following the referee's
suggestion
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