Semi-stable models for curves with cusps

Let R be a complete discrete valuation ring with residue characteristic zero, and let X be an integral regular flat curve over R with smooth generic fiber. Assume that the special fiber of X is smooth outside a single point where it has a cusp as singularity. We explicitly determine the structure of the minimal semi-stable model of X. In particular, we give an algebraic proof for the fact that the special fiber of any semi-stable model of X is treelike. This is equivalent to the finiteness of the monodromy of X over R. These two results were obtained in the 1970's by Lê Dung Tráng and A. Durfee using analytic methods. Received: 23 April 2001 / Published online: 8 November 2002 RID="*" ID="*" This research was made possible by the support of the Deutsche Forschungsgemeinschaft in form of grant LU 224/5-1.     

Semi-stable fibrations of generic p-rank 0

In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a filed of positive characteristic, if the p-rank of the generic fiber is 0, then the base change of the fibration by a sufficiently many iterative Frobenius morphism of the base curve violates the semi-positivity theorem. As an application, we suggest a statement on a distribution of p-ranks of reductions for a certain nonclosed point in the moduli space over a number field.     

The Picard–Lefschetz formula for p-adic cohomology

In this paper, we discuss a p-adic analogue of the Picard–Lefschetz formula. For a family with ordinary double points over a complete discrete valuation ring of mixed characteristic (0,p), we construct vanishing cycle modules which measure the difference between the rigid cohomology groups of the special fiber and the de Rham cohomology groups of the generic fiber. Furthermore, the monodromy operators on the de Rham cohomology groups of the generic fiber are described by the canonical generators of the vanishing cycle modules in the same way as in the case of the ℓ-adic (or classical) Picard–Lefschetz formula. For the construction and the proof, we use the logarithmic de Rham–Witt complexes and those weight filtrations investigated by Mokrane (Duke Math. J. 72(2):301–337, 1993).      

Trianalytic Subvarieties of the Hilbert Scheme of Points on a K3 Surface

Let X be a hyperk?hler manifold. Trianalytic subvarieties of X are subvarieties which are complex analytic with respect to all complex structures induced by the hyperk?hler structure. Given a K3 surface M, the Hilbert scheme classifying zero-dimensional subschemes of M admits a hyperk?hler structure. We show that for M generic, there are no trianalytic subvarieties of the Hilbert scheme. This implies that a generic deformation of the Hilbert scheme of K3 has no proper complex subvarieties. Submitted: May 1997, Revised version: December 1998     

Completions of UFDs with Semi-Local Formal Fibers

Let (T, M) be a complete local ring such that |T/M| = | T |. Given a finite set of incomparable nonmaximal prime ideals C of T, we provide necessary and sufficient conditions for T to be the completion of a local UFD A, whose generic formal fiber is semilocal with maximal ideals the elements of C. We also show that, given the T above, we can find necessary and sufficient conditions for T to be the completion of a UFD, whose formal fiber over a height one prime ideal is semilocal.

Communicated by I. Swanson.     

Efficient identity-based encryption with Hierarchical key-insulation from HIBE

Hierarchical key-insulated identity-based encryption (HKIBE) is identity-based encryption (IBE) that allows users to update their secret keys to achieve (hierarchical) key-exposure resilience, which is an important notion in practice. However, existing HKIBE constructions have limitations in efficiency: sizes of ciphertexts and secret keys depend on the hierarchical depth. In this paper, we first triumph over the barrier by proposing simple but effective design methodologies to construct efficient HKIBE schemes. First, we show a generic construction from any hierarchical IBE (HIBE) scheme that satisfies a special requirement, called MSK evaluatability introduced by Emura et al. (Des. Codes Cryptography 89(7):1535–1574, 2021). It provides several new and efficient instantiations since most pairing-based HIBE schemes satisfy the requirement. It is worth noting that it preserves all parameters’ sizes of the underlying HIBE scheme, and hence we obtain several efficient HKIBE schemes under the k-linear assumption in the standard model. Since MSK evaluatability is dedicated to pairing-based HIBE schemes, the first construction restricts pairing-based instantiations. To realize efficient instantiation from various assumptions, we next propose a generic construction of an HKIBE scheme from any plain HIBE scheme. It is based on Hanaoka et al.’s HKIBE scheme (Asiacrypt 2005), and does not need any special properties. Therefore, we obtain new efficient instantiations from various assumptions other than pairing-oriented ones. Though the sizes of secret keys and ciphertexts are larger than those of the first construction, it is more efficient than Hanaoka et al.’s scheme in the sense of the sizes of master public/secret keys.

     

The contact boundary of a complex polynomial

 We define the contact boundary of a complex polynomial f : ℂ ⁿ → ℂ as the intersection of some generic fiber with a large sphere. We show that, up to contact isotopy, this does not depend on the choice of the fiber (provided it is generic) and is invariant under polynomial automorphism of ℂ ⁿ . We next prove that the formal homotopy class of this contact boundary is invariant in a large family of deformations of polynomials, which are not necessarily topologically trivial. Received: 15 November 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 32S55, 53D15, 32S50     

Metrically Universal Generic Structures in Free Amalgamation Classes

We prove that each ∀₁ free amalgamation class K over a finite relational language L admits a countable generic structure M isometrically embedding all countable structuresin K relative to a fixed metric. We expand L by infinitely many binary predicates expressingdistance, and prove that the resulting expansion of K has a model companion axiomatizedby the first‐order theory of M. The model companion is non‐finitely axiomatizable, evenover a strong form of the axiom scheme of infinity.     

Discriminantal Arrangements, Fiber Polytopes and Formality

Manin and Schechtman defined the discriminantal arrangement of a generic hyperplane arrangement as a generalization of the braid arrangement. This paper shows their construction is dual to the fiber zonotope construction of Billera and Sturmfels, and thus makes sense even when the base arrangement is not generic. The hyperplanes, face lattices and intersection lattices of discriminantal arrangements are studied. The discriminantal arrangement over a generic arrangement is shown to be formal (and in some cases 3–formal), though it is in general not free. An example of a free discriminantal arrangement over a generic arrangement is given.     

Formal subgroups of abelian varieties

In this paper, we generalize the result of [12] in the following sense. Let A be an abelian variety over a number field k, let ? be the Néron model of A over the ring of integers O _k of k. Completing ? along its zero section defines a formal group over O _k . We prove that any formal subgroup of the generic fiber of whose closure in is smooth over an open subset of Spec O _k arises in fact from an abelian subvariety of A. The proof is of a transcendental nature and uses the Arakelovian formalism introduced by Bost [3]. Oblatum 2-V-2000 & 28-XI-2000?Published online: 5 March 2001     

Generic Fiber of Power Series Ring Extensions

Let D be a Noetherian domain containing a field, a ∈ D a nonzero nonunit, and z an indeterminate over D. We prove that the generic fiber of the extension D[[z]]? D[1/a][[z]] has dimension ≥ dim(D/aD).     

Generalized Divisors and Total Reflexivity

The notion of generalized divisors on schemes is introduced by Hartshorne. It is shown that there exists a bijection between the set of all generalized divisors on a scheme X and the set of all reflexive coherent 𝒪 _X -modules which are locally free of rank one at generic points. This bijection, corresponds Cartier divisors to the set of all locally free sheaves of rank one. Our aim in this article is to study the class of generalized divisors that maps to totally reflexive coherent 𝒪 _X -modules, under this correspondence. We investigate this class of divisors, that will be called Gorenstein divisors, both over schemes and also over commutative noetherian rings. We show that this class of divisors has usual properties and fits well in the hierarchy of divisors that already exists in the literature.     

Fiber-wise birational correspondences

We prove that each fiber-wise birational correspondence between smooth fibrations into (Fano) hypersurfaces, which is biregular on a generic fiber, is a fiber-wise isomorphism. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 120–130, July, 2000.     

Efficient hybrid encryption from ID-based encryption

This paper deals with generic transformations from ID-based key encapsulation mechanisms (IBKEM) to hybrid public-key encryption (PKE). The best generic transformation known until now is by Boneh and Katz and requires roughly 704-bit overhead in the ciphertext. We present new generic transformations that are applicable to partitioned IBKEMs. A partitioned IBKEM is an IBKEM that provides some extra structure. Such IBKEMs are quite natural and in fact nearly all known IBKEMs have this additional property. Our first transformation yields chosen-ciphertext secure PKE schemes from selective-ID secure partitioned IBKEMs with a 256-bit overhead in ciphertext size plus one extra exponentiation in encryption/decryption. As the central tool a Chameleon Hash function is used to map the identities. We also propose other methods to remove the use of Chameleon Hash, which may be of independent technical interest. Applying our transformations to existing IBKEMs we propose a number of novel PKE schemes with different trade-offs. In some concrete instantiations the Chameleon Hash can be made “implicit” which results in improved efficiency by eliminating the additional exponentiation. Since our transformations preserve the public verifiability property of the IBE schemes it is possible to extend our results to build threshold hybrid PKE schemes. We show an analogue generic transformation in the threshold setting and present a concrete scheme which results in the most efficient threshold PKE scheme in the standard model.     

Periodic Cyclic Homology as Sheaf Cohomology Dedicated to Daniel Quillen on his 60th Birthday

We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example, we show that infinitesimal hypercohomology with coefficients in K-theory gives the fiber of the Jones–Goodwillie character which goes from K-theory to negative cyclic homology.     

On the rank of the fibers of elliptic K3 surfaces

Let X be an elliptic K3 surface endowed with two distinct Jacobian elliptic fibrations π _i , i = 1, 2, defined over a number field k. We prove that there is an elliptic curve C ⊂ X such that the generic rank over k of X after a base extension by C is strictly larger than the generic rank of X. Moreover, if the generic rank of π _j is positive then there are infinitely many fibers of π _i (j ≠ i) with rank at least the generic rank of π _i plus one.     

Stable one-dimensional quasi-periodic cocycles on unitary group

In this paper, we study the stable one-dimensional quasi-periodic C ^∞ cocycles on U(N). We prove that any such cocycle on a generic irrational rotation is a limit point of reducible cocycles. The proof is based on Krikorian’s renormalization scheme and a local result of him.     

Generic and maximal Jordan types

For a finite group scheme G over a field k of characteristic p>0, we associate new invariants to a finite dimensional kG-module M. Namely, for each generic point of the projectivized cohomological variety we exhibit a “generic Jordan type” of M. In the very special case in which G=E is an elementary abelian p-group, our construction specializes to the non-trivial observation that the Jordan type obtained by restricting M via a generic cyclic shifted subgroup does not depend upon a choice of generators for E. Furthermore, we construct the non-maximal support variety Γ(G) _M , a closed subset of which is proper even when the dimension of M is not divisible by p.     

A uniqueness theorem for representations of GSO(6) and the strong multiplicity one theorem for generic representations of GSp(4)

Consider the θ-correspondence from GSp(4) to GSO(6). We prove that locally over a nonarchimedean fieldF, this correspondence is injective on generic representations (i.e. with Whittaker model) of GSp(4,F). We use this to show the strong multiplicity one property for irreducible, automorphic, cuspidal representations of GSp(4,A), which are generic.     

A stability condition for line bundles on reducible varieties

Over a family of varieties with singular special fiber, the relative Picard functor (i.e. the moduli space of line bundles) may fail to be compact. We propose a stability condition for line bundles on reducible varieties that is aimed at compactifying it. This stability condition generalizes the notion of ‘balanced multidegree’ used by Caporaso in compactifying the relative Picard functor over families of curves. Unlike the latter, it is defined ‘asymptotically’; an important theme of this paper is that although line bundles on higher-dimensional varieties are more complicated than those on curves, their behavior in terms of stability asymptotically approaches that of line bundles on curves.Using this definition of stability, we prove that over a one-parameter family of varieties having smooth total space, any line bundle on the generic fiber can be extended to a unique semistable line bundle on the (possibly reducible) special fiber, provided the special fiber is not too complicated in a combinatorial sense.