On Rational Surfaces Ruled by Conics

Abstract

We study projective rational surfaces ruled by conics, describing their singularities and special fibres. In particular, if Sis smooth, we give a “canonical” procedure to determine a minimal model among the geometrically ruled surfaces birational to S.     

Zur Theorie verallgemeinerter torsaler Strahlflächen

In this paper we will investigate (k+1)-dimensional generalized ruled surfaces generated by a one-parameter family ofk-dimensional linear subspaces of then-dimensional Euclidean spaceE _n . Some results which are well-known for developable surfaces are proved for generalized ruled surfaces: Generalized developable surfaces are locally either cyclinders, cones or tangent surfaces. Each regular surface on a generalized ruled surface is locally Euclidean if and only if is developable. Each locally Euclidean hypersurface is a generalized developable hypersurface. Furthermore, the hypersurfaces with vanishing Gaussian curvature and the locally Euclidean hypersurfaces on generalized rule hypersurfaces will be characterized.     

The Sp 3 -grassmannian and duality for prime Fano threefolds of genus 9

By a result of Mukai, the non-abelian Brill-Noether locus X=M_C(2, K : 3F) of type II, defined by a stable rank 2 vector bundle F of invariant 3 over a plane quartic curve C, is a prime Fano 3-fold X=X₁₆ of degree 16. The associate ruled surface S^X=P(F) is uniquely defined by X, and we see that for the general X=X₁₆, S^X is isomorphic to the Fano surface of conics on X. The argument uses the geometry of the Sp₃-grassmannian and the double projection from a line on X₁₆.Partially supported by Grant MM-1106/2001 of the Bulgarian Foundation for Scientific Research     

Lattice Points on Similar Figures and Conics

Let us say that a plane figure F satisfies Steinhaus’ condition if for any positive integer n, there exists a figure F _n similar to F which satisfies the condition |F_n?\mathbb Z²|=n{|F_n\cap{\mathbb Z}^2|=n}. For example, the circular disc satisfies Steinhaus’ condition. We prove that every compact convex region in the plane \mathbb R²{\mathbb R^2} satisfies Steinhaus’ condition. As for plane curves, it is known that the circle satisfies Steinhaus’ condition. We consider Steinhaus’ condition for other conics, and present several results.     

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c₁ = O_Q(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle O_Q(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c ₂(E) − 1 in \mathbb P₃^*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c ₂(E).     

False Hyperelliptic Surfaces with Section

Let X be a nonsingular relatively minimal projective surface over an algebraically closed field of characteristic p > 0. We call X a false hyperelliptic surface if X satisfies the following conditions: (1) c₂(X) = 0, c₁(X)² = 0, dim Alb (X) = 1, and (2) All fibres of the Albanese mapping of X are rational curves with only one cusp of type xp^v + yⁿ = 0. In this article, we consider a false hyperelliptic surface whose Albanese mapping has a cross-section. We prove that every false hyperellyptic surface with section arises from an elliptic ruled surface and that every false hyperelliptic surface has an elliptic fibration with multiple fibre. Moreover, we construct an example of false hyperelliptic surface with section, whose elliptic fibration has a multiple fibre of supersingular elliptic curve of multiplicity p^v (v > 1).     

Restricting semistable bundles on the projective plane to conics

We study the restrictions of rank 2 semistable vector bundles E on to conics. A Grauert-Mülich type theorem on the generic splitting is proven. The jumping conics are shown to have the scheme structure of a hypersurface of degree c₂(E) when c₁(E)=0 and of degree c₂(E)–1 when c₁(E)=–1. Some examples of jumping conics and jumping lines are studied in detail.Mathematics Subject Classification (2000):Primary:14J60; Secondary:14F05     

Inversion of degree <Emphasis Type="Italic">n</Emphasis> + 2

By the method of synthetic geometry, we define a seemingly new transformation of a three-dimensional projective space where the corresponding points lie on the rays of the first order, nth class congruence C _n ¹ and are conjugate with respect to a proper quadric Ψ. We prove that this transformation maps a straight line onto an n + 2 order space curve and a plane onto an n + 2 order surface which contains an n-ple (i.e. n-multiple) straight line. It is shown that in the Euclidean space the pedal surfaces of the congruences C _n ¹ can be obtained by this transformation. The analytical approach enables new visualizations of the resulting curves and surfaces with the program Mathematica. They are shown in four examples.      

The Subgroups of Direct Products of Surface Groups

A subgroup of a product of n surface groups is of type FP _n if and only if it contains a subgroup of finite index that is itself a product of (at most n) surface groups.     

Planar curve interpolation by piecewise conics of arbitrary type

Five points in general position inR ² always lie on a unique conic, and three points plus two tangents also have a unique interpolating conic, the type of which depends on the data. These well-known facts from projective geometry are generalized: an odd number 2n+1≥5 of points inR ², if they can be interpolated at all by a smooth curve with nonvanishing curvature, will have a uniqueGC ² interpolant consisting of pieces of conics of varying type. This interpolation process reproduces conics of arbitrary type and preserves strict convexity. Under weak additional assumptions its approximation order is ?(h ⁵), whereh is the maximal distance of adjacent data pointsf(t _i ) sampled from a smooth and regular planar curvef with nonvanishing curvature. Two algorithms for the construction of the interpolant are suggested, and some examples are presented.     

Sous-groupes discrets de <Emphasis Type="Italic">PU</Emphasis>(2,1) engendrés par <Emphasis Type="Italic">n</Emphasis> réflexions complexes et Déformation

Let PU(2,1) be the group of holomorphic isometries in the hyperbolic complex plane and let G _n be a sub-group of PU(2,1) which is generated by n complex reflections with respect to complex lines in . Under certain conditions, we prove that G _n is discrete. We construct representations ρ of the fundamental group Γ _g of the compact surface Σ _g of genus g, into PU(2,1), we prove they are discrete, faithful and we compute the dimension their deformation space.      

Eine liniengeometrische Deutung der Ebenen Gelenkvierecke

Let A_oB_o denote the base, A_oA and B_oB the arms of a plane four-bar linkage. It will be shown that the coupler-curve of A consists of a twice-counted circle k_a and the pair of isotropic lines through B_o. Similarly for B. The coupler-curve of any point at infinity is the line at infinity, six-fold counted. This and other properties are found by assuming k_a, k_b not in the same plane, but in different parallel planes. Then AB will describe a ruled surface, which is studied in the general case (grade 6, genus 1, the double curve consisting of 3 conics and 1 plane cubic, with 12 torsal lines) and in several special cases. As for general spatial four-bar linkages, remarks are given in 4.     

The Brauer group of an affine double plane associated to a hyperelliptic curve

We study the Brauer group of an affine double plane π:X→𝔸² defined by an equation of the form z² = f in two separate cases. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface ?¹×C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f = y²?p(x) is the equation of an affine hyperelliptic curve. For π as well as the unramified part of π, we compute the groups of divisor classes, the Brauer groups, the relative Brauer groups, and all of the terms in the sequences of Galois cohomology.     

On the notion of essential dimension for algebraic groups

We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. For example, if our groupG isS _n , these objects are field extensions; ifG=O _n , they are quadratic forms; ifG=PGL _n , they are division algebras (all of degreen); ifG=G ₂, they are octonion algebras; ifG=F ₄, they are exceptional Jordan algebras. We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the above-mentioned examples. In the last section we give an exposition of results, communicated to us by J.-P. Serre, relating essential dimension to Galois cohomology.Partially supported by NSA grant MDA904-9610022 and NSF grant DMS-9801675     

New Examples of Willmore Surfaces in S n

A surface x: M S ⁿ is called a Willmore surface if it is a criticalsurface of the Willmore functional _M (S – 2H ²)dv, where H isthe mean curvature and S is the square of the length of the secondfundamental form. It is well known that any minimal surface is aWillmore surface. The first nonminimal example of a flat Willmoresurface in higher codimension was obtained by Ejiri. This example whichcan be viewed as a tensor product immersion of S ¹(1) and a particularsmall circle in S ²(1), and therefore is contained in S ⁵(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in S ⁿ (1)which can be obtained as a tensor product immersion of two curves. We inparticular show that in this case too, one of the curves has to beS ¹(1), whereas the other one is contained either in S ²(1) or in S ³(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S ⁵. Also in the latter casewe explicitly include examples.     

Translation Lengths in Out(F n)

We prove that all elements of infinite order in Out(F _n) have positive translation lengths; moreover, they are bounded away from zero. As a consequence we get a new proof that solvable subgroups of Out(F _n) are finitely generated and virtually Abelian.     

Conjugacy in Permutation Representations of the Symmetric Group

Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e., that conjugacy classes of S _n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S _n  × S _n . We also consider the permutation representations of S _n which arise from the action of S _n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.     

A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

A short proof of the generalized Bonnet theorem

In three‐dimensional Euclidean space E³, the Bonnet theorem says that a curve on a ruled surface in three‐dimensional Euclidean space, having two of the following properties, has also a third one, namely, it can be a geodesic, that it can be the striction line, and that it cuts the generators under constant angle. In this work, in n dimensional Euclidean space Eⁿ, a short proof of the theorem generalized for (k + 1) dimensional ruled surfaces by Hagen in 4 is given. Copyright © 2013 John Wiley & Sons, Ltd.     

Laplace Invariants of Two-Dimensional Open Toda Lattices

We show that Toda lattices with the Cartan matrices A _n , B _n , C _n , and D _n are Liouville-type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized Laplace invariants. We show how they can be used to construct conservation laws (x and y integrals) and higher symmetries.