Cutting planes and the elementary closure in fixed dimension

The elementary closure $P'$ of a polyhedron $P$ is the intersection of $P$ with all its Gomory-Chvátal cutting planes. $P'$ is a rational polyhedron provided that $P$ is rational. The known bounds for the number of inequalities defining $P'$ are exponential, even in fixed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension. If $P$ is a simplicial cone, we construct a polytope $Q$, whose integral elements correspond to cutting planes of $P$. The vertices of the integer hull $Q_I$ include the facets of $P'$. A polynomial upper bound on their number can be obtained by applying a result of Cook et al. Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone that correspond to vertices of $Q_I$

Gomory-Chvátal cutting planes and the elementary closure of Polyhedra

The elementary closure P\u27; of a polyhedrom P is the intersection of P with all its Gomory-Chvátal cutting planes. P\u27; is a rational polyhedron provided that P is rational. The Chvátal-Gomory procedure is the iterative application of the elementary closure operation to P. The Chvátal rank is the minimal number of iterations needed to obtain P_I. It is always finite, but already in |R² one can construct polytopes of arbitrary large Chvátal rank. We show that the Chvátal rank of polytopes contained in the n-dimensional 0/1 cube is O(n² log n) and prove the lower bound (1+E) n, for some E> 0. We show that the separation problem for the elementary closure of a rational polyhedron is NP-hard. This solves a problem posed by Schrijver. Last we consider the elementary closure in fixed dimension. the known bounds for the number of inequalities defining P\u27; are exponential, even fixed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension. Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone from this polynomial description in fixed dimension with a maximal degree of violation in a natural sense.Die elemtare Hülle P\u27; eines Polyeders P ist der Durchschnitt von P mit all seinen Gomory-Chvátal Schnittebenen. P\u27; ist ein rationales Polyeder, falls P rational ist. Die Chvátal-Gomory Prozedur ist das wiederholte Bilden der elementaren Hülle, beginnend mit P. Die minimale Anzahl der Iterationen, die bis zum Erhalt der ganzzahligen Hülle P1 von P nötig sind, heißt Chvátal-Rang von P. Der Chvátal-Rang eines rationalen Polyeders ist endlich. Jedoch lassen sich bereits im |R² Beispiele mit beliebig hohem Chvátal-Rang konstruieren. Wir zeigen, dass der Chvátal-Rang eines Polytops im n-dimensionalen 0/1 Würfel durch O(n² log n) beschränkt ist, und beweisen die untere Schranke (1 + epsilon) * n, für ein epsilon > 0. Wir zeigen, dass das Separationsproblem für die elementare Hülle eines rationalen Polyeders NP-hart ist. Dies löst ein von Schrijver formuliertes Problem. Schließlich wenden wir uns der elementaren Hülle rationaler Polyeder in fester Dimension zu. Die bislang bekannten Schranken für die Anzahl der Ungleichungen, die zur Darstellung von P\u27; benötigt werden, sind exponentiell, selbst in fester Dimension. Wir zeigen, dass in fester Dimension P\u27; durch polynomiell viele Ungleichungen beschrieben werden kann. Wir entwerfen außerdem einen, in beliebiger Dimension polynomiellen, Algorithmus, der zu einem spitzen Kegel P eine Schnittebene aus der polynomiellen Darstellung von P\u27; berechnet, die zudem einen maximalen Grad der Verletzung in einem natürlichen Sinne aufweist

Some extremal contractions between smooth varieties arising from projective geometry

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective n-dimensional varieties, n\geq 4, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed. The examples considered come from particular special homaloidal and subhomaloidal linear systems, which usually are degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron. The first series of examples is associated to particular codimension 2 determinantal smooth subvarieties of P^m, 3\leq m\leq 5. We get another series of examples by considering special cubic hypersurfaces through some surfaces in P^5, or some 3-folds in P^7 having one apparent double point. The last examples come from an intriguing birational elementary extremal contraction in dimension 6, studied by Semple and Tyrrell and fully described in the last section.Comment: 29 pages. Proc. London Math. Society, to appea

Fake Real Planes: exotic affine algebraic models of R2\mathbb{R}^{2}

We study real rational models of the euclidean affine plane $\mathbb{R}^{2}$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective plane $\mathbb{R}\mathbb{P}^{2}$ is well known: up to birational diffeomorphisms, $\mathbb{P}^{2}(\mathbb{R})$ is the only model. A fake real plane is a smooth geometrically integral surface $S$ defined over $\mathbb{R}$ not isomorphic to $\mathbb{A}^2_\mathbb{R}$, whose real locus $S(\mathbb{R})$ is diffeomorphic to $\mathbb{R}^2$ and such that the complex surface $S_\mathbb{C}(\mathbb{C})$ has the rational homology type of $\mathbb{A}^2_\mathbb{C}$. We prove that fake planes exist by giving many examples and we tackle the question: does there exist fake planes $S$ such that $S(\mathbb{R})$ is not birationally diffeomorphic to $\mathbb{A}^2_\mathbb{R}(\mathbb{R})$?Comment: 36 pages, 18 figure

Some Loci of Rational Cubic Fourfolds

In this paper we investigate the divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$, whose generic element is a smooth cubic containing a smooth quartic scroll. Using the fact that all degenerations of quartic scrolls in $\mathbb P^5$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we conclude that every cubic hypersurface belonging to $\mathcal C_{14}$ is rational. As an application of our results and of the construction of some explicit examples contained in the Appendix, we also prove that the Pfaffian locus is not open in $\mathcal C_{14}$.Comment: 24 pages. Final, rewritten and expanded version with some new results, to appear on Math. Annale

Deformations and stability in complex hyperbolic geometry

This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sullivan. Despite Goldman-Millson-Yue rigidity results for such complex manifolds of infinite volume, we present different classes of such manifolds that allow non-trivial (quasi-Fuchsian) deformations and point out that such flexible manifolds have a common feature being Stein spaces. While deformations of complex surfaces from our first class are induced by quasiconformal homeomorphisms, non-rigid complex surfaces (homotopy equivalent to their complex analytic submanifolds) from another class are quasiconformally unstable, but nevertheless their deformations are induced by homeomorphisms