Existence of a non‐reflexive embedding with birational Gauss map for a projective variety

We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙ^N to re‐embed into some projective space ℙ^M so as to be non‐reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙ^N is identically zero; hence the projective varietyX re‐embedded in ℙ^M yields a negative answer to Kleiman–Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙ^N with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙ^N satisfying the assumption. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The injectivity of the extended Gauss map of general projections of smooth projective varieties

Let X be a smooth n-dimensional projective variety embedded in some projective space ℙ ^N over the field ℂ of the complex numbers. Associated with the general projection of X to a space ℙ ^N-m (N-m>n+1) one defines an extended Gauss map (in case N-m>2n-1 this is the Gauss map of the image of X under the projection). We prove that is smooth. In case any two different points of X do have disjoint tangent spaces then we prove that is injective.     

Preservation of the injectivity of the Gauss map for general projections

Let X be a smooth irreducible quasi-projective variety of dimension n in P ^N with N ≥ 2n + 2. Let γ be its Gauss map, let be the embedding obtained from the general projection in P ^N and let γ′ be its Gauss map. We say that the general projection preserves the injectivity of the Gauss map if γ(Q) ≠ γ(Q′) implies γ′(Q) ≠ γ′ (Q′). We prove that this property holds in the following cases: N≥ 3n + 1; N ≥ 3n with n ≥ 2; N ≥ 3n−1 with n ≥ 4 and X does not contain a linear (n−1)-space. In case N = 3n−1 and X does contain a linear (n−1)-space (such smooth varieties exist) then the general projection does not preserver the injectivity of the Gauss map. This shows that there does not exist a straightforward kind of Bertini theorem for properties related to the Gauss map. The author is affiliated with the University at Leuven as a research fellow. This paper belongs to the FWO-project G.0318.06.     

On the Structure of Submanifolds with Degenerate Gauss Maps

An n-dimensional submanifold X of a projective space P ^N (C) is called tangentially degenerate if the rank of its Gauss mapping gamma;; X G(n, N) satisfies 0 < rank < n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space P ^N (C). By means of the focal images, three basic types of submanifolds are discovered: cones, tangentially degenerate hypersurfaces, and torsal submanifolds. Moreover, for tangentially degenerate submanifolds, a structural theorem is proven. By this theorem, tangentially degenerate submanifolds that do not belong to one of the basic types are foliated into submanifolds of basic types. In the proof the authors introduce irreducible, reducible, and completely reducible tangentially degenerate submanifolds. It is found that cones and tangentially degenerate hypersurfaces are irreducible, and torsal submanifolds are completely reducible while all other tangentially degenerate submanifolds not belonging to basic types are reducible.     

ON MULTIOSCULATING SPACES

The aim of this paper is to study varieties with second Gauss map not birational. In particular we classify such varieties in dimension two. We prove that there are two types of surfaces S of P ⁿ (C), with n > 5, not satisfying Laplace equations, with second Gauss map t ² not birational: i. surfaces such that the image of the second Gauss map is one-dimensional and containing a one-dimensional family of curves. Each curve of the family is contained in some P ³ ? P ⁿ .

ii. surfaces such that the second Gauss map is generically finite of degree at least two. In this case the image of the second Gauss map is two-dimensional, locally embedded in a Laplace congruence and meeting the general generatrix in more than one point.

     

A Filtration on the Chow Groups of a Complex Projective Variety

Let X/ _C be a projective algebraic manifold, and further let CH ^k (X) _Q be the Chow group of codimension k algebraic cycles on X, modulo rational equivalence. By considering Q-spreads of cycles on X and the corresponding cycle map into absolute Hodge cohomology, we construct a filtration {F ^l}_{l 0} on CH ^k (X) _Q of Bloch-Beilinson type. In the event that a certain conjecture of Jannsen holds (related to the Bloch-Beilinson conjecture on the injectivity, modulo torsion, of the Abel–Jacobi map for smooth proper varieties over Q), this filtration truncates. In particular, his conjecture implies that F ^k+1 = 0.     

Dually Degenerate Varieties and the Generalization of a Theorem of Griffiths–Harris

The dual variety X^* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X^* is a hypersurface in the dual space (P^N)^*. If dimX^*<N–1, then the variety X is called dually degenerate. The authors refine these definitions for a variety XP^N with a degenerate Gauss map of rankr. For such a variety, in the general case, the dimension of its dual variety X^* is N–l–1, where l=n–r, and X is dually degenerate if dimX^*<N–l–1. In 1979 Griffiths and Harris proved that a smooth variety XP^N is dually degenerate if and only if all its second fundamental forms are singular. The authors generalize this theorem for a variety XP^N with a degenerate Gauss map of rankr. Mathematics Subject Classification (2000) 53A20.     

Harmonic maps in unfashionable geometries

 We describe some general constructions on a real smooth projective 4-quadric which provide analogues of the Willmore functional and conformal Gauss map in Lie sphere and projective differential geometry. Extrema of these functionals are characterized by harmonicity of this Gauss map. Received: 3 August 2001     

The separability of the Gauss map versus the reflexivity

In projective algebraic geometry, various pathological phenomena in positive characteristic have been observed by several authors. Many of those phenomena concerning the behavior of embedded tangent spaces seem to be controlled by the separability of (the extension of function fields defined by) the Gauss map, or by the reflexivity with respect to the projective dual for a projective variety. The purpose of this paper is to survey the studies on the relationship between the separability of the Gauss map and the reflexivity for a projective variety: Is the separability of the Gauss map equivalent to the reflexivity for a projective variety?     

Groups acting on finite dimensional spaces with finite stabilizers

It is shown that every H -group G of type admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a -module which is projective as a -module for all finite then M has finite projective dimension. Received: April 30, 1997     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

Any algebraic variety in positive characteristic admits a projective model with an inseparable Gauss map

We determine the values attained by the rank of the Gauss map of a projective model for a fixed algebraic variety in positive characteristic p. In particular, it is shown that any variety in p>0 has a projective model such that the differential of the Gauss map is identically zero. On the other hand, we prove that there exists a product of two or more projective spaces admitting an embedding into a projective space such that the differential of the Gauss map is identically zero if and only if p=2.     

On the projectively almost-factorial varieties

Summary A projectively normal subvariety (X,O _X) ofP ^N(k), k an algebraically closed field of characteristic zero, will be said to be projectively almost-factorial if each Weil divisor has a multiple which is a complete intersection in X. The main result is the following: (X,O _X) is projectively almost-factorial if and only if for all x ∈ X the local ringO _x is almost-factorial and the quotient ofPic(X) modulo the subgroup generated by the class ofO _X (1) is torsion. We also prove the invariance of the projective almost-factoriality up to isomorphisms and state some relations between the projective almost-factoriality (resp. projective factoriality) of X and the almost-factoriality (resp. factoriality) of the affine open subvarieties. Finally we discuss some consequences of the main result in the case k=ℂ: in particular we prove that the Picard group of a projectively almost-factorial variety is isomorphic to the Néron-Severi group, hence finitely generated. Entrata in Redazione il 23 aprile 1976. AMS(MOS) subject classification (1970): Primary 14C20, 13F15.     

Mumford coverings of the projective line

A Mumford covering of the projective line over a complete non-archimedean valued field K is a Galois covering X? P¹_K X\rightarrow {\bf P}^1_K such that X is a Mumford curve over K. The question which finite groups do occur as Galois group is answered in this paper. This result is extended to the case where P¹_K {\bf P}^1_K is replaced by any Mumford curve over K.     

The singular locus of the secant varieties of a smooth projective curve

Let X be a smooth irreducible non-degenerated projective curve in some projective space P^N. Let r be a positive integer such that 2r + 1 < N and let S_r(X) be the r-th secant variety of X. It is a variety of dimension 2r + 1. In this paper we prove that the singular locus is the (r - 1)-th secant variety S_{r- 1}(X) if X does not have any (2r + 2)-secant 2r-space divisor. Received: 26 November 2002     

Threefolds with big and nef anticanonical bundles II

In a follow-up to our paper [Threefolds with big and nef anticanonical bundles I, Math. Ann., 2005, 333(3), 569–631], we classify smooth complex projective threefolds Xwith −K _X big and nef but not ample, Picard number γ(X) = 2, and whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop X ⁺ are not both birational.     

An affirmative answer to a question of Ciliberto

We give an example of a nondegeneraten-dimensional smooth projective varietyX inP ²ⁿ⁺¹ with the canonical bundle ample a varietyX whose tangent variety TanX has dimension less than 2n over an algebraically closed field of any characteristic whenn≥9. This varietyX is not ruled by lines and the embedded tangent space at a general point ofX intersectsX at some other points, so that this yields an affirmative answer to a question of Ciliberto.     

Projective Valuations,D-Relations,and Ends of Λ-Trees

Abstract

A projective valuation on a set Eis a mapping w : E ⁴ → Λ ∪ {±∞}, where Λ is an ordered abelian group, satisfying certain axioms. A D-relation on Eis a four-place relation on E, again with certain properties. There is a projective valuation on the set of ends of a Λ-tree (and on any subset, by restriction) and we show, using a construction suggested by Tits in the case Λ = ?, that every projective valuation arises in this way. Every projective valuation wdefines a D-relation, and there is a simple geometric interpretation of the D-relation, given a Λ-tree defining w. Our main result is a converse, that any D-relation can be defined by a projective valuation, hence arises from an embedding into the set of ends of a Λ-tree.     

On the Dilogarithmic Cycle Class Map

For a smooth projective surface X over C, we construct uncountably many non-torsion cycles in CH²(X) which die in the dilogarithmic cohomology of S. Bloch whenever there is an Abelian variety A and a correspondence δ in CH²(X × A) which induces non-zero map on the spaces of global 2-forms. In case X = E × E with E an elliptic curve, all of albanese kernel dies in any such analytic cohomology. Similar results are obtained for higher dimensional varieties under the condition of existence of non-trivial decomposable 2-forms.