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Let S be a smooth projective surface over the complex numbers and let c1 ∈ H2 (S, Z) and c2 ∈ H4 (S, Z) be two classes. For an ample divisor H on S, one can study the moduli space MH(c1, c2) of H-semistable torsion-free sheaves E on S of rank 2 with c1(E) = c1 and c2(E) = c2. We want to study the change of MH(c1, c2) under variation of H. It is known that the ample cone of

All given rings in this paper are commutative, associative with identity, and Noetherian. Recently, L. Ein, R. Lazarsfeld, and K. Smith [ELS] discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if h is the largest height of an associated prime of I, then I (hn) ⊆ In for all n ≥ 0. Here, if W is the complement of the union of the associated primes of

Abstract. The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism ϕ: R→S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R-modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R. A third is to obtain criteria for a ring equipped with a ‘contracting ’ endomorphism—such as the Frobenius endomorphism—to be regular or complete intersection; these results represent broad generalizations of Kunz’s characterization of regularity in prime characteristic.

Abstract. Results on L-functions of Artin-Schreier coverings by Dwork, Bombieri and Adolphson-Sperber are generalized to L-functions of Witt coverings. Key words: L-functions, exponential sums, Newton polygon MSC2000: 11L07, 14F30 1

1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of

Abstract. The generic Newton polygons for L-functions of exponential sums associated to Laurent polynomials in one variable are determined when p is large. The corresponding Hasse polynomials are also determined. Key words: exponential sum, L-function, generic Newton polygon MSC2000: 11L07, 14F30 1.

Abstract. In this paper we prove that the spherical subalgebra eH1,τ e of the double affine Hecke algebra H1,τ is an integral Cohen-Macaulay algebra isomorphic to the center Z of H1,τ,andH1,τ e is a Cohen-Macaulay eH1,τ e-module with the property H1,τ =EndeH1,τ e(H1,τ e) whenτisnot a root of unity. In the case of the root system An−1 the variety Spec(Z) issmoothand coincides with the completion of the configuration space of the Ruijenaars-Schneider system. It implies that the module eH1,τ is projective and all irreducible finite dimensional representations of H1,τ are isomorphic to the regular representation of the finite Hecke algebra.

Abstract. Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an antinilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the Ascending Chain Condition. 1.

To the memory of G. Chogoshvili Abstract. Let R be a local ring and A a connected differential graded algebra over R which is free as a graded R-module. Using homological perturbation theory techniques, we construct a minimal free multi model for A having properties similar to that of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute ‘multi ’ refers to the category of multicomplexes. 2000 Mathematics Subject Classification. 18G10, 18G35, 18G55, 55P35, 55P62, 55U15, 57T30. Key words and phrases. Models for differential graded algebras, minimal models for differential graded algebras over local rings, multicomplex, multialgebra, homological perturbations. 2 JOHANNES HUEBSCHMANN