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17
Multigraded Hilbert schemes
 J. Algebraic Geom
"... 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, includ ..."
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Cited by 48 (2 self)
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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.
A better upper bound on the number of triangulations of a planar point set
 Journal of Combinatorial Theory, Ser. A
"... Abstract. We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 n O(n −6) straightedge triangulations of its convex hull. This improves the previous upper bound of 276.75 n+O(log(n)). ..."
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Cited by 30 (3 self)
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Abstract. We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 n O(n −6) straightedge triangulations of its convex hull. This improves the previous upper bound of 276.75 n+O(log(n)).
The Generalized Baues Problem For Cyclic Polytopes
, 1998
"... 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 di ..."
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Cited by 11 (6 self)
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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 nontrivial 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 nonregular 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 nonregular for every choice of points on the moment curve.
The Graph of Triangulations of a Point Configuration With D+4 Vertices is 3Connected.
, 1999
"... 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 3connected for acyclic vector configurations, which include all point ..."
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Cited by 10 (6 self)
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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 3connected 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 ...
Combinatorics Of The Toric Hilbert Scheme
 DISCRETE COMPUT. GEOM
, 2002
"... The toric Hilbert scheme is a parameter space for all ideals with the same multigraded 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 t ..."
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Cited by 8 (5 self)
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The toric Hilbert scheme is a parameter space for all ideals with the same multigraded 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
Generalized integer partitions, tilings of zonotopes and lattices
 Proceedings of the 12th international conference Formal Power Series and Algebraic Combinatorics (FPSAC'00
, 2000
"... Abstract: In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple dynamics, have the distributive lattice structure. Likewi ..."
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Cited by 7 (4 self)
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Abstract: In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, 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 flips and dynamical systems involving partitions.
On the Refinements of a Polyhedral Subdivision
 COLLECT. MATH
, 2000
"... 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 a ..."
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Cited by 7 (4 self)
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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 nonregular subdivisions then it has nonregular triangulations and we provide simple proofs of the existence of nonregular subdivisions for many particular point configurations.
Rhombus tilings: decomposition and space structure
, 2004
"... We study the spaces of rhombus tilings, i.e. the graphs whose vertices are tilings of a fixed zonotope, and two tilings are linked if one can pass from one to the other one by a local transformation, called flip. We first use a decomposition method to encode rhombus tilings and give a useful charact ..."
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Cited by 3 (2 self)
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We study the spaces of rhombus tilings, i.e. the graphs whose vertices are tilings of a fixed zonotope, and two tilings are linked if one can pass from one to the other one by a local transformation, called flip. We first use a decomposition method to encode rhombus tilings and give a useful characterization for a sequence of bits to encode a tiling. In codimension 2, we use the previous coding to get a canonical representation of tilings, and an order structure on the space of tilings, which is shown to be a graded poset, from which connectivity is deduced.
On The Baues Conjecture in corank 3.
, 2000
"... 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 polyhedr ..."
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Cited by 2 (0 self)
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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...