Given a lattice polytope Q ⊆ IR n , we define an affine scheme ¯ M 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 ¯ M 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.
We show that Gorenstein singularities that are cones over singular Fano varieties provided by so-called flag quivers are smoothable in codimension three. Moreover, we give a precise characterization about the smoothability in codimension three of the Fano variety itself.
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… (More)
The obstruction space T 2 and the cup product T 1 ×T 1 → T 2 are computed for toric singularities.
There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring K[x 1 ,. .. , x n ]. 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 sub-graphs on multigraded Hilbert schemes and on… (More)
Simplicial complexes X provide commutative rings A(X) via the Stanley-Reisner 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.
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.