The Projective Geometry of the Gale Transform

The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises, in subjects as diverse as optimization, coding theory, theta-functions, and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture. In this paper we reexamine the Gale transform in the light of modern algebraic geometry. We give a more general definition, in the context of finite (locally) Gorenstein subschemes. We put in modern form a number of the more remarkable examples discovered in the past, and we add new constructions and connections to other areas of algebraic geometry. We generalize Goppa's theorem in coding theory and we give new applications to Castelnuovo theory. We give references to classical and modern sources.Comment: 43 pages, Plain Tex, uses diagrams.tex. Postscript file also available from http://www.math.columbia.edu/~psori

Gale transform of a starshaped sphere

Gale transform is a simple but powerful tool in convex geometry. In particular, the use of Gale transform is the main argument in the classification of polytopes with few vertices. Many books and documents cover the definition of Gale transform and its main properties related to convex polytopes. But it seems that there does not exist document studying the Gale transform of more general objects, such that triangulation of spheres. In this paper, we study the properties of the Gale transform of a large class of such spheres called starshaped spheres.Comment: 21 pages. Affine and convex geometry; combinatorics; Gale transfor

Colorful simplicial depth, Minkowski sums, and generalized Gale transforms

The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to prove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. (2006). Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton (2003) in the theory of normal surfaces.Comment: 17 pages, 3 figure

Gale Duality and Free Resolutions of Ideals of Points

What is the shape of the free resolution of the ideal of a general set of points in P^r? This question is central to the programme of connecting the geometry of point sets in projective space with the structure of the free resolutions of their ideals. There is a lower bound for the resolution computable from the (known) Hilbert function, and it seemed natural to conjecture that this lower bound would be achieved. This is the ``Minimal Resolution Conjecture'' (Lorenzini [1987], [1993]). Hirschowitz and Simpson [1994] showed that the conjecture holds when the number of points is large compared with r, but three examples (with r = 6,7,8) discovered computationally by Schreyer in 1993 show that the conjecture fails in general. We describe a novel structure inside the free resolution of a set of points which accounts for the observed failures and provides a counterexample in P^r for every r\geq 6, r\neq 9. The geometry behind our construction occurs not in P^r but in a different projective space, in which there is a related set of points, the ``Gale transform'' (or ``associated set'', in the sense of Coble.)Comment: 25 pages, plain TeX, author-supplied DVI file available at http://www.math.columbia.edu/~psorin/eprints/gale.dvi . This version fixes a mathematical error in the end of section

Equal coefficients and tolerance in coloured Tverberg partitions

The coloured Tverberg theorem was conjectured by B\'ar\'any, Lov\'{a}sz and F\"uredi and asks whether for any d+1 sets (considered as colour classes) of k points each in R^d there is a partition of them into k colourful sets whose convex hulls intersect. This is known when d=1,2 or k+1 is prime. In this paper we show that (k-1)d+1 colour classes are necessary and sufficient if the coefficients in the convex combination in the colourful sets are required to be the same in each class. We also examine what happens if we want the convex hulls of the colourful sets to intersect even if we remove any r of the colour classes. Namely, if we have (r+1)(k-1)d+1 colour classes of k point each, there is a partition of them into k colourful sets such that they intersect using the same coefficients regardless of which r colour classes are removed. We also investigate the relation of the case k=2 and the Gale transform, obtaining a variation of the coloured Radon theorem

Rectilinear Crossings in Complete Balanced d-Partite d-Uniform Hypergraphs

In this paper, we study the embedding of a complete balanced $d$-partite $d$-uniform hypergraph with all its $nd$ vertices represented as points in general position in $\mathbb{R}^d$ and each hyperedge drawn as a convex hull of $d$ corresponding vertices. We assume that the set of vertices is partitioned into $d$ disjoint sets, each of size $n$, such that each of the vertices in a hyperedge is from a different set. Two hyperedges are said to be crossing if they are vertex disjoint and contain a common point in their relative interiors. Using the Generalized Colored Tverberg Theorem, we observe that such an embedding of a complete balanced $d$-partite $d$-uniform hypergraph with $nd$ vertices contains $\Omega\left((8/3)^{d/2}\right){\left({n/2}\right)^d{\left((n-1)/2\right)}^d}$ crossing pairs of hyperedges for sufficiently large $n$ and $d$. Using the Gale Transform and the Ham-Sandwich Theorem, we improve this lower bound to $\Omega\left(2^{d}\right){\left({n/2}\right)^d{\left((n-1)/2\right)}^d}$ for sufficiently large $n$ and $d$

Conformal blocks and rational normal curves

We prove that the Chow quotient parametrizing configurations of n points in $\mathbb{P}^d$ which generically lie on a rational normal curve is isomorphic to $\overline{M}_{0,n}$, generalizing the well-known $d = 1$ result of Kapranov. In particular, $\overline{M}_{0,n}$ admits birational morphisms to all the corresponding geometric invariant theory (GIT) quotients. For symmetric linearizations the polarization on each GIT quotient pulls back to a divisor that spans the same extremal ray in the symmetric nef cone of $\overline{M}_{0,n}$ as a conformal blocks line bundle. A symmetry in conformal blocks implies a duality of point-configurations that comes from Gale duality and generalizes a result of Goppa in algebraic coding theory. In a suitable sense, $\overline{M}_{0,2m}$ is fixed pointwise by the Gale transform when $d=m-1$ so stable curves correspond to self-associated configurations.Comment: 17 pages, 1 figure; published versio

Commuting difference operators and the combinatorial Gale transform

We study the spectral theory of $n$-periodic strictly triangular difference operators $L=T^{-k-1}+\sum_{j=1}^k a_i^j T^{-j}$ and the spectral theory of the "superperiodic" operators for which all solutions of the equation $(L+1)\psi=0$ are (anti)periodic. We show that for a superperiodic operator $L$ there exists a unique superperiodic operator ${\cal L}$ of order $(n-k-1)$ which commutes with $L$ and show that the duality $L\leftrightarrow {\cal L}$ coincides up to a certain involution with the combinatorial Gale transform recently introduced in [21].Comment: 19 pages, Late

Exterior algebra methods for the Minimal Resolution Conjecture

If r\geq 6, r\neq 9, we show that the Minimal Resolution Conjecture fails for a general set of m points in P^r for almost 1/2\sqrt r values of m. This strengthens the result of Eisenbud and Popescu [1999], who found a unique such m for each r in the given range. Our proof begins like a variation of that of Eisenbud and Popescu, but uses exterior algebra methods as explained by Eisenbud and Schreyer [2000] to avoid the degeneration arguments that were the most difficult part of the Eisenbud-Popescu proof. Analogous techniques show that the Minimal Resolution Conjecture fails for linearly normal curves of degree d and genus g when d\geq 3g-2, g\geq 4, reproving results of Schreyer, Green, and Lazarsfeld.Comment: 15 pages, Plain TeX, uses diagrams.te

Unstable hyperplanes for Steiner bundles and multidimensional matrices

We study some properties of the natural action of $SL(V_0) \times...\times SL(V_p)$ on nondegenerate multidimensional complex matrices $A\in\P (V_0\otimes...\otimes V_p)$ of boundary format(in the sense of Gelfand, Kapranov and Zelevinsky); in particular we characterize the non stable ones,as the matrices which are in the orbit of a "triangular" matrix, and the matrices with a stabilizer containing \C^*, as those which are in the orbit of a "diagonal" matrix. For $p=2$ it turns out that a non degenerate matrix $A\in\P (V_0\otimes V_1\otimes V_2)$ detects a Steiner bundle $S_A$ (in the sense of Dolgachev and Kapranov) on the projective space $\P^{n}, n = dim (V_2)-1$. As a consequence we prove that the symmetry group of a Steiner bundle is contained in SL(2) and that the SL(2)-invariant Steiner bundles are exactly the bundles introduced by Schwarzenberger [Schw], which correspond to "identity" matrices. We can characterize the points of the moduli space of Steiner bundles which are stable for the action of $Aut(\P^n)$, answering in the first nontrivial case a question posed by Simpson. In the opposite direction we obtain some results about Steiner bundles which imply properties of matrices. For example the number of unstable hyperplanes of $S_A$ (counting multiplicities) produces an interesting discrete invariant of $A$, which can take the values $0, 1, 2, ... ,\dim~V_0+1$ or $\infty$; the $\infty$ case occurs if and only if $S_A$ is Schwarzenberger (and $A$ is an identity). Finally, the Gale transform for Steiner bundles introduced by Dolgachev and Kapranov under the classical name of association can be understood in this setting as the transposition operator on multidimensional matrices.Comment: 27 pages, plain tex, a missing case in theorem 2.6 is adde