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Journals and Conferences
Given a lattice polytope Q ⊆ IR, we define an affine scheme M̄ that reflects the possibilities of splitting Q into a Minkowski sum. On the other hand, Q induces a toric Gorenstein singularity Y , and we construct a flat family over M̄ with Y as special fiber. In case Y has an isolated singularity only, this family is versal.
We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our theory extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.
(1.1) The break through in deformation theory of (two-dimensional) quotient singularities Y was Kollár/Shepherd-Barron’s discovery of the one-to-one correspondence between so-called P-resolutions, on the one hand, and components of the versal base space, on the other hand (cf. [KS], Theorem (3.9)). It generalizes the fact that all deformations admitting a… (More)
Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a “proper polyhedral divisor” introduced in earlier work, we develop the concept of a “divisorial fan” and show that these objects encode the equivariant gluing of… (More)
BACKGROUND The effects of neuromuscular electrical stimulation (NMES) in critically ill patients after cardiothoracic surgery are unknown. The objectives were to investigate whether NMES prevents loss of muscle layer thickness (MLT) and strength and to observe the time variation of MLT and strength from preoperative day to hospital discharge. METHODS In… (More)
There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring K[x1, . . . , xn]. The definition involves Gröbner bases or the action of the algebraic torus (K∗)n. We present algorithms for computing the (affine schemes representing) edges in this graph. We study the induced subgraphs on multigraded Hilbert schemes and on… (More)
Simplicial complexes X provide commutative rings A(X) via the StanleyReisner construction. We calculated the cotangent cohomology, i.e., T 1 and T 2 of A(X) in terms of X. These modules provide information about the deformation theory of the algebro geometric objects assigned to X.
(1.1) Let σ be a rational, polyhedral cone. It induces a (normal) affine toric variety Yσ which may have singularities. We would like to investigate its deformation theory. The vector space T 1 Y of infinitesimal deformations is multigraded, and its homogeneous pieces can be determined by combinatorial formulas developed in [Al 1]. If Yσ only has an… (More)
We investigate how the chain property for the associated primes of monomial degenerations of toric (or lattice) ideals can be generalized to arbitrary A-graded ideals. The generalization works in dimension d = 2, but it fails for d ≥ 3.
We prove a vanishing theorem for the Hodge number h of projective toric varieties provided by a certain class of polytopes. We explain how this Hodge number also gives information about the deformation theory of the toric Gorenstein singularity derived from the same polytope. In particular, the vanishing theorem for h implies that these deformations are… (More)