A topological proof of cancellation theorem for ${\vec C}^2$

We give a new proof of the Cancellation Theorem for which says that if V is a smooth complex affine surface such that is isomorphic to then V is isomorphic to The proof is based on the topological method of Mumford-Ramanujam. Received: 2 November 2000 / in final form: 10 September 2001 / Published online: 1 February 2002     

Generic polynomials are descent-generic

Let be a generic polynomial for a group G in the sense that every Galois extension N/L of infinite fields with group G and K≤L is given by a specialization of g(X). We prove that then also every Galois extension whose group is a subgroup of G is given in this way. Received: 15 January 2001     

On quasi-amorphous sets

A set is said to be amorphous if it is infinite, but cannot be written as the disjoint union of two infinite sets. The possible structures which an amorphous set can carry were discussed in [5]. Here we study an analogous notion at the next level up, that is to say replacing finite/infinite by countable/uncountable, saying that a set is quasi-amorphous if it is uncountable, but is not the disjoint union of two uncountable sets, and every infinite subset has a countably infinite subset. We use the Fraenkel–Mostowski method to give many examples showing the diverse structures which can arise as quasi-amorphous sets, for instance carrying a projective geometry, or a linear ordering, or both; reconstruction results in the style of [1] are harder to come by in this case. Received: 8 April 1999 / Published online: 3 October 2001     

Special points on products of two Shimura curves

Let S ₁ and S ₂ be two Shimura curves over ℚ attached to rational indefinite quaternion algebras B _{1 ℚ} and B _{1 ℚ} with maximal orders B ₁ and B ₂ respectively. We consider an irreducible closed algebraic curve C in the product (S ₁×S ₂)_ℂ such that C(ℂ) ∩ (S ₁×S ₂)(ℂ) contains infinitely many complex multiplication points. We prove, assuming the Generalized Riemann Hypothesis (GRH) for imaginary quadratic fields, that C is of Hodge type. Received: 3 January 2000 / Revised version: 2 October 2000     

Bounded Positive Solutions of Schrödinger Equations in Two-Dimensional Exterior Domains

We prove under quite general assumptions the existence of a bounded positive solution to the semilinear Schrödinger equation in a two-dimensional exterior domain.     

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.     

Testing sets for properness of polynomial mappings

In this paper we characterize testing sets for properness of polynomial mappings . We also study the set of points at which such mappings are not proper. As the first application we give a proof of the Complementary Conjecture of McKay and Wang (Conjecture 12 in [16]). The second application is an answer to the problem of Kraft- Russell about a characterization of The third application is (given in [13]) a solution of the Problem of Van den Essen and Shpilrain about endomorphism of the ring The fourth application is a theorem about extensions of affine varieties. Received: 27 February 1998 / in revised form: 10 January 1999     

A note on log etale Čech cohomolog

In this paper, we prove the comparison theorem between log etale cohomology and log etale Čech cohomology for certain log schemes, which is a generalization of a result of Artin on the comparison theorem between etale cohomology and etale Čech cohomology. It turns out that the naive generalization is not true, and we also give a counter-example for it. Received: 2 July 2000 / Revised version: 4 September 2000     

When Does a Submodule of (ℝ[x 1 ⋯, x k]) n Contain a Positive Element?

We characterize for modules consisting of tuples of Laurent polynomials with real coefficients whether such a module contains a positive element. The two conditions needed are numerical and directional positivity. The proof applies universal Gröbner bases.     

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S. Received: 28 March 2000 / Revised version: 20 October 2000     

Algebraic loop groups and moduli spaces of bundles

We study algebraic loop groups and affine Grassmannians in positive characteristic. The main results are normality of Schubert-varieties, the construction of line-bundles on the affine Grassmannian, and the proof that they induce line-bundles on the moduli-stack of torsors. Received March 15, 2001 / final version received April 11, 2002?Published online November 19, 2002     

Schottky uniformization¶and vector bundles over Riemann surfaces

We study a natural map from representations of a free group of rank g in GL(n,ℂ), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X. Maximally unstable flat bundles are shown to arise in this way. We give a necessary and sufficient condition for this map to be a submersion, when restricted to representations producing stable bundles. Using a generalized version of Riemann's bilinear relations, this condition is shown to be true on the subspace of unitary Schottky representations. Received: 13 June 2000 / Revised version: 29 December 2000     

Projective resolutions associated to projections

In this paper we will describe projective resolutions of d dimensional Cohen–Macaulay spaces X by means of a projection of X to a hypersurface in d+1-dimensional space. We will show that for a certain class of projections, the resulting resolution is minimal. Received: 22 February 1999     

Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains

Consider the nonstationary Stokes equations in exterior domains with the compact boundary . We show first that the solution decays like for all as . This decay rate is optimal in the sense that for some as occurs if and only if the net force exerted by the fluid on is zero. Received: 15 June 2000 / Published online: 18 June 2001     

Epimorphic extensions and Prüfer extensions¶ of partially ordered rings

The paper contributes to the investigation of epimorphisms in the category of reduced partially ordered rings (porings). Two main questions are considered: 1) Does the set of isomorphism classes of a given poring have a largest element (an epimorphic hull)? 2) Given an epimorphic extension, or even a Prüfer extension, f:A→B of porings: how closely are A and B related to each other? Received: 17 June 1999     

On connectedness of sets in the real spectra of polynomial rings

Let R be a real closed field. The Pierce–Birkhoff conjecture says that any piecewise polynomial function f on R ⁿ can be obtained from the polynomial ring R[x ₁,..., x _n ] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points , the existence of connected sets in the real spectrum of R[x ₁,..., x _n ], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce–Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce–Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce–Birkhoff conjecture in dimension greater than 2. Thirdly, we apply these ideas to give two proofs that the Connectedness conjecture (and hence also the Pierce–Birkhoff conjecture in the abstract formulation) holds locally at any pair of points , one of which is monomial. One of the proofs is elementary while the other consists in deducing this result as an immediate corollary of the main connectedness theorem of this paper.     

A bound on the Castelnuovo-Mumford regularity for curves

Recall that a projective curve in with ideal sheaf is said to be n-regular if for every integer and that in this case, it is cut out scheme-theoretically by equations of degree at most n. The purpose here is to show that an irreducible, reduced, projective curve of degree d and large arithmetic genus satisfies a smaller regularity bound than the optimal one . For example, if then a curve is -regular unless it is embedded by a complete linear system of degree . Received: 29 May 2000 / Published online: 24 September 2001     

Schubert polynomials of types A--D

Schubert polynomials of type B, C, and D have been described first by S. Billey and M. Haiman [BH] using a combinatorial method. In this paper we give a unified algebraic treatment of Schubert polynomials of types A–D in the style of the Lascoux–Schützenberger theory in type A, i.e. Schubert polynomials are generated by the application of sequences of divided difference operators to “top polynomials”. The use of the creation operators for Q-Schur and P-Schur functions allows us to give: (1) simple and natural forms of the “top polynomials”, (2) formulas for the easy computation with all divided differences, (3) recursive structures, and (4) simplified derivations of basic properties. Received: 23 July 1998     

The Nash sheaf of a complete resolution

Every three-dimensional complex algebraic variety with isolated singular point has a resolution factoring through the Nash blowup and the blowup of the maximal ideal over which the second Fitting ideal sheaf is locally principal. In such resolutions one can construct Hsiang–Pati coordinates and thus obtain generators for the Nash sheaf that are the differentials of monomial functions. Received: 9 April 2001 / Revised version: 26 July 2001