We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, including Bayer’s conjecture on equations defining Grothendieck’s classical Hilbert scheme and the construction of a Chow morphism for toric Hilbert schemes. 1.

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We study the graph of bistellar flips between triangulations of a vector configuration A with d + 4 elements in rank d + 1 (i.e. with corank 3), as a step in the Baues problem. We prove that the graph is connected in general and 3-connected for acyclic vector configurations, which include all point configurations of dimension d with d + 4 elements. Hence, every pair of triangulations can be joined by a finite sequence of bistellar flips and every triangulation has at least 3 geometric bistellar neighbours. In corank 4, connectivity is not known and having at least 4 flips is false. In corank 2, the results are trivial since the graph is a cycle. Our methods are based in a dualization of the concept of triangulation of a point or vector configuration A to that of virtual chamber of its Gale transform B, introduced by de Loera et al. in 1996. As an additional result we prove a topological representation theorem for virtual chambers, stating that every virtual chamber of a rank 3 vector ...

The Generalized Baues Problem asks whether for a given point configuration the order complex of all its proper polyhedral subdivisions, partially ordered by refinement, is homotopy equivalent to a sphere. In this paper, an affirmative answer is given for the vertex sets of cyclic polytopes in all dimensions. This yields the first non-trivial class of point configurations with neither a bound on the dimension, the codimension, nor the number of vertices for which this is known to be true. Moreover, it is shown that all triangulations of cyclic polytopes are lifting triangulations. This contrasts the fact that in general there are many non-regular triangulations of cyclic polytopes. Beyond this, we find triangulations of C 11 5 with flip deficiency. This proves—among other things—that there are triangulations of cyclic polytopes that are non-regular for every choice of points on the moment curve.

: In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of 2D-gons (hexagons, octagons, decagons, ...), using dynamical systems and order theory. We show that the sets of partitions, ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of zonotopes, ordered with a simple and classical dynamics, is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical systems exist. These results give a better understanding of the behaviour of tilings of zonotopes with ips and dynamical systems involving partitions. Keywords : Integer partitions, Tilings of Zonotopes, Random tilings, Lattices, Sand Pile Model, Discrete Dynamical Systems. 1 Preliminaries In this paper, we mainly deal with two kinds of combinatorial objects: integer partitions and tilings. An integer partition problem is a set of...

Let : P ! Q be an affine projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by (or -induced) and the fiber polytope of the projection: a polytope \Sigma(P; ) of dimension dim(P ) \Gamma dim(Q) whose faces are in correspondence with the coherent -induced subdivisions (or -coherent subdivisions). In this paper we investigate the structure of the poset of -induced refinements of a -induced subdivision. In particular, we define the refinement polytope associated to any -induced subdivision S, which is a generalization of the fiber polytope and shares most of its properties. As applications of the theory we prove that if a point configuration has non-regular subdivisions then it has non-regular triangulations and we provide simple proofs of the existence of non-regular subdivisions for many particular point configurations.

The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P 4 whose associated toric Hilbert schemes have arbitrary dimension. We show that

Sets of monomial ideals arise in combinatorics and computational algebra in several different ways. The first such collection considered is the set of all initial ideals of a given ideal in a commutative polynomial ring. Initial ideals, which arise in Gr obner basis theory, enable us to transform questions about the ideal to simpler questions about the related monomial ideal. A universal finiteness theorem for sets of monomial ideals is given, one of whose consequences is a new proof that the set of initial ideals of an ideal is finite. We also study the term orders which give rise to initial ideals, relating them to the Baues problem. Specializing to toric ideals, we then consider the intersection of all initial ideals, called the vertex ideal. Another way a set of monomial ideals arises is as the torus-fixed points of the toric Hilbert scheme. We construct a graph whose vertices are these monomial ideals, which is connected if and only if the scheme is connected. We also give a map from this graph to the Baues graph of triangulations of the associated point configuration.

The cyclic zonotope Z…n; d † is the zonotope in R d generated by any ndistinct vectors of the form …1; t; t2;...; td 1 †. It is proved that the refinement poset of all proper zonotopal subdivisions of Z…n; d † which are induced by the canonical projection p: Z…n; d0 †!Z…n; d†, in the sense of Billera and Sturmfels, is homotopy equivalent to a sphere and that any zonotopal subdivision of Z…n; d † is shellable. The first statement gives an affirmative answer to the generalized Baues problem in a new special case and refines a theorem of Sturmfels and Ziegler on the extension space of an alternating oriented matroid. An important ingredient in the proofs is the fact that all zonotopal subdivisions of Z…n; d † are stackable in a suitable direction. It is shown that, in general, a zonotopal subdivision is stackable in a given direction if and only if a certain associated oriented matroid program is Euclidean, in the sense of Edmonds and Mandel.

A special case of the Generalized Baues Conjecture states that the order complex of the Baues poset of an acyclic vector configuration A (the Baues complex of A) is homotopy equivalent to a sphere of dimension equal to the corank of A minus one. The Baues poset of A is the set of proper polyhedral subdivisions of A ordered by refinement. Recently, Santos has found a counterexample in corank 317 to the Baues conjecture. Here, we study the case of corank 3. The techniques we use also allow us to show that if a corank 3 vector configuration is not acyclic, then its Baues complex is contractible. Introduction The Baues problem concerns the study of the space of the polyhedral subdivisions of a vector configuration [15]. A vector configuration A in R d+1 is a finite spanning set of labelled vectors (we allow repetitions) in the linear space R d+1 . A cell of A is any spanning subset of A. The number d + 1 is called the rank of A, while the difference between the cardinality of...