171 research outputs found
Sweeping the cd-Index and the Toric h-Vector
We derive formulas for the cd-index and the toric h-vector of a convex
polytope P from a sweeping by a hyperplane. These arise from interpreting the
corresponding S-shelling of the dual of P. We describe a partition of the faces
of the complete truncation of P to reflect explicitly the nonnegativity of its
cd-index and what its components are counting. One corollary is a quick way to
compute the toric h-vector directly from the cd-index. We also propose an
"extended toric" h-vector that fully captures the information in the flag
h-vector.Comment: 23 page
FLAG \u3cem\u3eF\u3c/em\u3e-VECTORS OF POLYTOPES WITH FEW VERTICES
We may describe a polytope P as the convex hull of n points in space. Here we consider the numbers of chains of faces of P. The toric g-vector and CD-index of P are useful invariants for encoding this information. For a simplicial polytope P, Lee defined the winding number wk in a Gale diagram corresponding to P. He showed that wk in the Gale diagram equals gk of the corresponding polytope. In this dissertation, we fully establish how to compute the g-vector for any polytope with few vertices from its Gale diagram. Further, we extend these results to polytopes with higher dimensional Gale diagrams in certain cases, including the case when all the points are in affinely general position. In the Generalized Lower Bound Conjecture, McMullen and Walkup predicted that if gk(P)=0 for some simplicial polytope P and some k, then P is (k-1)-stacked. Lee and Welzl independently use Gale transforms to prove the GLBC holds for any simplicial polytope with few vertices. In the context of Gale transforms, we will extend this result to nonpyramids with few vertices. We will also prove how to obtain the CD-index of polytopes dual to polytopes with few vertices in several cases. For instance, we show how to compute the CD-index of a polytope from the Gale diagram of its dual polytope when the Gale diagram is 2-dimensional and the origin is captured by a line segment
Polytopes, Hopf algebras and Quasi-symmetric functions
In this paper we use the technique of Hopf algebras and quasi-symmetric
functions to study the combinatorial polytopes. Consider the free abelian group
generated by all combinatorial polytopes. There are two natural
bilinear operations on this group defined by a direct product and a
join of polytopes. is a commutative
associative bigraded ring of polynomials, and is a commutative associative
threegraded ring of polynomials. The ring has the structure of a
graded Hopf algebra. It turns out that has a natural Hopf
comodule structure over . Faces operators that send a
polytope to the sum of all its -dimensional faces define on both rings
the Hopf module structures over the universal Leibnitz-Hopf algebra
. This structure gives a ring homomorphism \R\to\Qs\otimes\R,
where is or . Composing this homomorphism with
the characters of , of
, and with the counit we obtain the ring homomorphisms
f\colon\mathcal{P}\to\Qs[\alpha],
f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha], and
\F^*:\mathcal{RP}\to\Qs, where is the Ehrenborg transformation. We
describe the images of these homomorphisms in terms of functional equations,
prove that these images are rings of polynomials over , and find the
relations between the images, the homomorphisms and the Hopf comodule
structures. For each homomorphism , and \F the images
of two polytopes coincide if and only if they have equal flag -vectors.
Therefore algebraic structures on the images give the information about flag
-vectors of polytopes.Comment: 61 page
Cox rings of degree one del Pezzo surfaces
Let X be a del Pezzo surface of degree one over an algebraically closed field
(of any characteristic), and let Cox(X) be its total coordinate ring. We prove
the missing case of a conjecture of Batyrev and Popov, which states that Cox(X)
is a quadratic algebra. We use a complex of vector spaces whose homology
determines part of the structure of the minimal free Pic(X)-graded resolution
of Cox(X) over a polynomial ring. We show that sufficiently many Betti numbers
of this minimal free resolution vanish to establish the conjecture.Comment: Significant revision. Proposition 4.4 fixes gap in previous version.
To appear in Algebra and Number Theor
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