10.2 Polytopes and their connection to toric geometry
4 min read•july 30, 2024
Polytopes are geometric objects that generalize polygons and polyhedra to higher dimensions. They're crucial in toric geometry, serving as a bridge between combinatorics and algebraic geometry. Their structure encodes important information about toric varieties.
The of a polytope corresponds to the orbit stratification of its associated toric variety. This connection allows us to study complex algebraic varieties using simpler combinatorial objects, revealing deep insights into both fields.
Polytopes and their properties
Definition and basic concepts
Top images from around the web for Definition and basic concepts
mg.metric geometry - Convex hull on a Riemannian manifold - MathOverflow View original
Is this image relevant?
Politopo convexo - Wikipedia, la enciclopedia libre View original
Is this image relevant?
Icositetrahemiicositetrachoron - Polytope Wiki View original
Is this image relevant?
mg.metric geometry - Convex hull on a Riemannian manifold - MathOverflow View original
Is this image relevant?
Politopo convexo - Wikipedia, la enciclopedia libre View original
Is this image relevant?
1 of 3
Top images from around the web for Definition and basic concepts
mg.metric geometry - Convex hull on a Riemannian manifold - MathOverflow View original
Is this image relevant?
Politopo convexo - Wikipedia, la enciclopedia libre View original
Is this image relevant?
Icositetrahemiicositetrachoron - Polytope Wiki View original
Is this image relevant?
mg.metric geometry - Convex hull on a Riemannian manifold - MathOverflow View original
Is this image relevant?
Politopo convexo - Wikipedia, la enciclopedia libre View original
Is this image relevant?
1 of 3
- A polytope is a generalization of polygons and polyhedra to higher dimensions defined as the convex hull of a finite set of points in a real vector space
- The dimension of a polytope is the dimension of its affine hull, which is the smallest affine subspace containing the polytope
- Faces of a polytope are its intersections with supporting hyperplanes, and they form a partially ordered set (poset) under inclusion, called the face lattice
- Vertices, edges, and facets are faces of dimensions 0, 1, and d-1, respectively, where d is the dimension of the polytope
Combinatorial properties
- The of a polytope counts the number of faces of each dimension, while the is a transformation of the f-vector with combinatorial significance
- A polytope is simplicial if all its proper faces are simplices, and it is simple if each is incident to exactly d edges
- The of a is a graded algebra encoding its face numbers, and it is isomorphic to the cohomology ring of the associated toric variety
- Combinatorial operations on polytopes, such as and Minkowski sums, have corresponding operations on toric varieties (projective duality and fiber products)
- The g-theorem characterizes the possible f-vectors of simplicial polytopes and has deep connections to the cohomology of toric varieties and the hard Lefschetz theorem
- The generalized Dehn-Sommerville relations express linear dependencies among the face numbers of a simplicial polytope, reflecting the Poincaré duality of the cohomology ring of the associated toric variety
Polytopes vs Toric Varieties
Correspondence between polytopes and toric varieties
- Toric varieties are algebraic varieties that contain a torus as a dense open subset, with a natural action of the torus on the variety
- Every rational polytope gives rise to a , where the polytope encodes the combinatorial data of the variety
- The face lattice of the polytope corresponds to the orbit stratification of the toric variety under the torus action
- The vertices of the polytope correspond to the fixed points of the torus action on the variety, and the edges correspond to the invariant curves connecting these fixed points
- The dimension of the polytope equals the complex dimension of the associated toric variety
- Smooth polytopes (those where the primitive edge directions at each vertex form a basis) correspond to smooth toric varieties
Constructing toric varieties from polytopes and vice versa
- Given a rational polytope, the associated toric variety can be constructed as a geometric quotient of an open subset of affine space by a torus action determined by the polytope
- The homogeneous coordinate ring () of the toric variety is a polynomial ring with variables corresponding to the lattice points of the polytope, quotiented by an ideal determined by the combinatorics of the polytope
- Conversely, given a projective toric variety, one can recover the associated polytope as the convex hull of the lattice points corresponding to the generators of the homogeneous coordinate ring
- The of the polytope encodes the torus orbits on the variety and their closures, with cones corresponding to torus-invariant affine open subsets
- between toric varieties correspond to lattice maps between the associated polytopes that respect the face structures
Geometric and combinatorial invariants of polytopes
Volume and lattice point enumeration
- The volume of a polytope can be computed using the Euclidean volume of its faces, with alternating signs based on the face dimensions ()
- The of a rational polytope counts the number of lattice points in dilations of the polytope
- Its coefficients encode geometric information about the polytope and the associated toric variety
Cohomological invariants
- The Stanley-Reisner ring of a simplicial polytope is isomorphic to the cohomology ring of the associated toric variety
- The g-theorem characterizes the possible f-vectors of simplicial polytopes and has connections to the cohomology of toric varieties and the hard Lefschetz theorem
- The generalized Dehn-Sommerville relations express linear dependencies among the face numbers of a simplicial polytope, reflecting the Poincaré duality of the cohomology ring of the associated toric variety
Key Terms to Review (25)
B. Sturmfels: B. Sturmfels is a prominent mathematician known for his significant contributions to algebraic geometry, particularly in the context of toric varieties and polytopes. His work often intersects the fields of combinatorics, optimization, and algebra, establishing deep connections between geometric structures and algebraic properties. Sturmfels' research has advanced the understanding of how polytopes can be used to model algebraic objects, particularly in toric geometry.
Convex Polytope: A convex polytope is a geometric object defined as the convex hull of a finite set of points in a Euclidean space, often represented as a bounded intersection of half-spaces. This structure can be thought of as a higher-dimensional generalization of polygons and polyhedra, and it plays a crucial role in various mathematical fields, including optimization and combinatorial geometry, particularly in the context of toric geometry.
Cox Ring: The Cox ring is a specific kind of graded ring associated with a toric variety, capturing the algebraic and geometric structure of the variety in a unified way. This ring is built from the homogeneous coordinate rings of the affine patches of a toric variety and plays a crucial role in studying the properties and computations related to these varieties, including their intersection theory and their relations to polytopes.
E. Miller: E. Miller is known for significant contributions to the study of polytopes and their applications in toric geometry. His work explores the intricate relationships between polytopes, algebraic varieties, and combinatorial structures, making a substantial impact on how these elements interact in the realm of geometry. Through his research, Miller helped advance the understanding of how polytopes can be utilized to study properties of toric varieties and their associated algebraic structures.
Ehrhart Polynomial: The Ehrhart polynomial is a mathematical expression that counts the number of integer points in the dilated version of a polytope. Specifically, for a rational polytope, this polynomial provides a way to understand how the volume of integer lattice points changes as the polytope is scaled. It connects to various concepts in combinatorial geometry and algebraic geometry, especially in relation to toric varieties and their properties.
Euler-Poincaré Formula: The Euler-Poincaré formula relates the topology of a convex polytope to its combinatorial structure. Specifically, it states that for a convex polytope, the Euler characteristic is equal to the number of vertices minus the number of edges plus the number of faces, represented mathematically as $$ ext{V} - ext{E} + ext{F} = ext{χ}$$. This formula provides a foundational connection between algebraic geometry and combinatorics, highlighting how the geometric properties of polytopes can be analyzed through their vertex-edge-face relationships.
F-vector: An f-vector is a vector that encapsulates the combinatorial data of a convex polytope, specifically counting the number of faces of each dimension. The entries of the f-vector correspond to the number of vertices, edges, faces, and higher-dimensional features of the polytope. This concept is closely tied to the study of polytopes and plays a significant role in toric geometry, as it helps relate geometric properties to algebraic structures.
Face Lattice: The face lattice of a polytope is a combinatorial structure that organizes its faces (vertices, edges, and higher-dimensional faces) into a hierarchy based on inclusion. Each face of the polytope corresponds to a node in the lattice, and there is a directed edge from one face to another if the first is a face of the second. This organization helps in understanding the relationships among the faces and their dimensions, linking polytopes to various concepts in toric geometry.
Facet: A facet is a flat side or face of a polytope, which is a higher-dimensional geometric figure. In the context of polytopes, facets are the faces of dimension one less than the polytope itself; for instance, in a three-dimensional polytope like a cube, the facets are the two-dimensional faces or squares. Understanding facets is crucial for studying properties such as volume, surface area, and connections to toric geometry, as they serve as building blocks for more complex structures.
Fan: In mathematics, a fan is a collection of cones that are used to describe the combinatorial structure of toric varieties. Each cone represents a direction in a multi-dimensional space, and the way these cones intersect defines the geometry of the associated toric variety. Fans provide a systematic way to study the properties of toric varieties, connecting algebraic geometry and combinatorial geometry through their underlying polytopes and facilitating computations in toric geometry.
Gorenstein Criterion: The Gorenstein Criterion is a set of conditions used to determine whether a certain type of algebraic variety, specifically those defined by homogeneous ideals, possesses the Gorenstein property. This property indicates that the variety has particularly nice geometric and algebraic features, such as dualizing modules being free and singularities being mild. Understanding this criterion helps in connecting polytopes to toric varieties, as it provides a bridge between combinatorial properties of polytopes and the geometric properties of associated algebraic varieties.
H-vector: The h-vector is a polynomial invariant that encodes combinatorial information about a simplicial polytope, particularly its face numbers. This concept is crucial in understanding the properties of polytopes and their connections to toric varieties, providing insight into the geometry and topology of these structures through the lens of algebraic geometry.
Klein's Theorem: Klein's Theorem states that every convex polytope can be realized as a compact subset of a Euclidean space, which means that there exists a geometric representation of the polytope in a way that preserves its combinatorial structure. This theorem connects the algebraic properties of polytopes to their geometric representations, making it significant in both algebraic geometry and combinatorial theory.
Normal Fan: A normal fan is a collection of cones in a vector space that corresponds to the faces of a polytope, capturing the combinatorial and geometric properties of that polytope. This structure helps in understanding how polytopes relate to toric varieties, as each cone in the normal fan reflects the way that the facets of the polytope connect to points in the associated toric variety.
Polar Duality: Polar duality is a geometric principle that establishes a correspondence between points and hyperplanes in projective geometry. This concept provides a powerful tool in studying the properties of polytopes and toric varieties, allowing for the exploration of dual relationships between geometric objects and their associated algebraic structures.
Polar Polytope: A polar polytope is a geometric structure associated with a given polytope, capturing its dual properties. Specifically, it is formed by taking the convex hull of the linear functionals that correspond to the vertices of the original polytope, thus linking its algebraic features to its geometric representation in toric geometry. This relationship allows for a deeper understanding of the interplay between the properties of polytopes and their duals, enriching the study of toric varieties.
Polyhedral Decomposition: Polyhedral decomposition refers to the process of breaking down a complex polyhedron into simpler components, typically convex polytopes. This technique is crucial in computational geometry and plays an important role in various applications, such as optimization problems, robot motion planning, and toric geometry. Understanding how to decompose these structures allows mathematicians and computer scientists to analyze properties and relationships between them effectively.
Projective Toric Variety: A projective toric variety is a specific type of algebraic variety that is defined by combinatorial data from a convex polytope and can be embedded into projective space. These varieties provide a bridge between algebraic geometry and combinatorial geometry, allowing for the study of geometric properties using the underlying combinatorial structure of polytopes.
Simplicial Polytope: A simplicial polytope is a type of polytope where all its faces are simplices, which are the simplest possible polytopes in any given dimension. Each vertex of a simplicial polytope connects to a set of edges forming triangles or higher-dimensional analogs. These structures play a crucial role in combinatorial geometry and provide a bridge to toric geometry, where they can represent both geometric and algebraic objects through their associated fan and torus actions.
Smooth Toric Variety: A smooth toric variety is a type of algebraic variety that can be associated with a fan of cones in a vector space, where the variety is defined by monomials corresponding to the lattice points of a convex polytope. These varieties are characterized by their well-behaved geometric properties, such as having no singularities, which makes them particularly useful in both algebraic geometry and combinatorics. The connection to polytopes comes from the fact that every smooth toric variety can be realized as the geometric realization of a fan, where each cone corresponds to a facet of the polytope.
Stanley-Reisner Ring: The Stanley-Reisner ring is a commutative ring associated with a simplicial complex, constructed using the polynomial ring in variables corresponding to the vertices of the complex, modulo the ideal generated by certain monomials. This ring captures important algebraic and combinatorial information about the simplicial complex, linking geometry and algebra through toric varieties and providing insights into properties like homology and cohomology.
Support Function: The support function is a mathematical tool that describes how a polytope interacts with its supporting hyperplanes. It provides a way to express the maximum value of a linear functional on the points of the polytope, revealing important geometric properties and relationships in toric geometry. This function plays a crucial role in understanding polytopes' structure and their connection to convex analysis.
Toric Ideal: A toric ideal is a specific type of ideal in a polynomial ring that is generated by binomials corresponding to the relations among monomials associated with a rational polytope. These ideals are crucial in understanding the combinatorial and geometric properties of toric varieties, which are algebraic varieties constructed from combinatorial data related to convex polytopes. They reveal deep connections between algebra, geometry, and combinatorics.
Toric Morphisms: Toric morphisms are maps between toric varieties that arise from the combinatorial data of fans and polytopes. These morphisms are defined through the relationships between the corresponding cones in the fans, preserving the algebraic structure of the varieties involved. They play a crucial role in understanding how different toric varieties relate to one another and how geometric properties can be derived from the underlying combinatorial data.
Vertex: A vertex is a fundamental point in geometry, defined as a corner or a meeting point of lines or edges. In the context of polytopes, vertices serve as the defining elements that help form the shape and structure of the polytope. The arrangement and connection of these vertices define various properties, such as dimension, volume, and the relationship to toric geometry, where vertices correspond to lattice points that play a crucial role in the study of toric varieties.