We construct an elliptic analogue of Sklyanin‘s separation of variables for the sl(2) Gaudin system, using an adaptation of Drinfeld‘s Radon transformations.

This paper extends to quasi-projective varieties earlier work by the author and H. Blaine Lawson concerning spaces of algebraic cocycles on projective varieties. The topological monoid $C_r(Y)(U)$ of effective cocycles on a normal, quasi-projective variety $U$ with values in a projective variety $Y$ consists of algebraic cycles on $U×Y$ equi-dimensional of relative dimension r over $U$. A careful choice of topology enables the establishment of various good properties: the definition is essentially algebraic, the group completion $Z_r(Y)(U)$ has “sensible” homotopy groups, the construction is contravariant with respect to $U$, convariant with respect to $Y$, and there is a natural “quality map” to the topological group of cycles on $U×Y$. The fundamental theorem presented here is the extension of Friedlander-Lawson duality to this context: the duality map $Z_r(Y)(U)$ to $Z_r+m(U × Y)$ is a homotopy equivalence provided that both $U$ and $Y$ are smooth (where $m=dim U$). Various application are given, especially the determination of the homotopy types of certain topological groups of algeb raic morphisms.

For an extension R [rarrhk] A of commutative Noetherian rings the behavior of the associated morphism of topological spaces Spec A → Spec R is often measured by its behavior on each of its fibers. Specifically, one studies the ’splitting‘ (or ’branching‘) and the ’ramification‘ that occurs in each fiber. In the classical constructions of faithfully flat analytic extensions (e.g., completion or Henselization) of excellent local rings the splitting and ramification properties are fairly well understood; see EGA IV [6, 18.10], Nagata [13, Sect. 37] or Raynaud [15, Ch. IX]. The strongest results are usually achieved for fibers over a ’normal point‘ of Spec R, that is, over p ∈ Spec R such that R/p is a normal domain [e.g., the property of a normal prime p in a local ring to be ’unibranched‘, i.e., the Henselization of R/p is a (normal) domain].

This paper studies the second pluri-genus of surface singularities. We give a formula for this invariant of a Gorenstein singularity, and several inequalities relating the invariant with the Milnor number, Tjurina number and the modality of a hypersurface singularity.

Each finite dimensional irreducible rational representation V of the symplectic group Sp$_2g$(Q) determines a generically defined local system V over the moduli space M$_g$ of genus g smooth projective curves. We study H$^2$ (M$_g$; V) and the mixed Hodge structure on it. Specifically, we prove that if g [ges ] 6, then the natural map IH$^2$(M˜$_g$; V) → H$^2$(M$_g$; V) is an isomorphism where M˜$_g$ is the Satake compactification of M$_g$. Using the work of Saito we conclude that the mixed Hodge structure on H$^2$(M$_g$; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H$^2$(M$_g$; V) for 3 [les ] g < 6. Results of this article can be applied in the study of relations in the Torelli group T$_g$.

We consider the moduli space $M$ of stable principal $G$-bundles over a compact Riemann surface $C$ of genus $g ⩾ 2, G$ being any reductive algebraic group and give an explicit description of the generic fibre of the Hitchin map $H:T^*M→ K$. If $T ⊂ G$ is a fixed maximal torus with Weyl group $W$, for each given generic element $&phis; ∈ K$ one may construct a $W$-Galois covering $&Ctilde;$ of $C$ and consider the generalized Prym variety $P=Hom_W(X(T),J(&Ctilde;))$, where $X(T)$ denotes the group of characters on $T$ and $J(&Ctilde;)$ the Jacobian. The connected component $P_0 ⊂ P$ which contains the trivial element is an abelian variety. In the present paper we use the classical theory of representations of finite groups to compute dim $P = dim M$. Next, by means of mostly elementary techniques, we explicitly construct a finite map $F$ from each connected component $H^-1(&phis;)_c$ of the Hitchin fibre to $P_0$ and study its degree. In case $G=PGl(2)$ one has that the generic fibre of $F:H^-1$()$_c → P_0$ is a principal homogeneous space with respect to a product of $(2d-2)$ copies of $Z/2Z$ where $d$ is the degree of the canonical bundle over $C$. However if the Dynkin diagram of $G$ does not contain components of type $B_l, l⩾ 1$ or when the commutator subgroup $(G,G)$ is simply connected the map $F$ is injective.

Let X be an open subset of C$^n$ and (f$_1$, …,f$_p$): X → C$^p$ be a holomorphic mapping. We prove that if (x$^0$,0, λ$^0$) ∈ T$^*$ × C$^p$ does not belong to the characteristic variety of the D$_X$ [λ]-module D$_X$[λ]f$^&lgr;$, then there exists a conic neighborhood V × Γ of (x$^0$, λ$^0$) such the function (λ$_1$, …, λ$_p$) [map ] ∫ | f$_1$ |$^lgr_1$ … | f$_p$ | $^lgr_p$ ω is rapidely decreasing in | Im λ | for λ ∈ Γ with Re λ bounded, for any (n,n)-form ω of class C$^∞$ with compact support in V. The following partial converse of this result is also established: if s [map ] ∫$_f=s$ φ is of class C$^∞$ in C$^p$ for all (n,n)-forms φ of class C$^∞$ with compact support in X, then d f$_1$ ∧ … ∧ d f$_p$ (x) ≠ 0, ∀ x ∈ X.

Let A and x be d × d and d × 1 real matrices. We study the asymptotic distribution of the sequence ((A$^n$x) mod.Z$^d$) in the torus $T^d$. We prove that for any A, for almost all x, this distribution exists; we characterize the case where this distribution is uniform; we give a description of the non uniform case. Finally we ask a question on the asymptotic distribution modulo 1 of the coefficients of the sequence of powers (A$^n$).

We study here the O-dimension Chow group of some particular smooth varieties which are fibrations over curves (i.e. are equipped with a proper, surjective morphism over a smooth curve). We obtain finiteness results for fibrations, whose generic fibres are Severi-Brauer varieties of squarefree index, when the group field is a number field (or in some particular cases a finitely generated field over Q).

This article treats compactifications of the space of generalized spin curves. Generalized spin curves, or r-spin curves, are pairs (X,L) with X a smooth curve and L a line bundle whose rth tensor power is isomorphic to the canonical bundle of X. These are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), which have been of interest recently, in part because of their applications to fermionic string theory. Three different compactifications over Z[1/r], all using torsion-free sheaves, are constructed. All three yield algebraic stacks, one of which is shown to have Gorenstein singularities that can be described explicitly, and one of which is smooth. All three compactifications generalize constructions of Deligne and Cornalba done for the case when r=2.

For a real Enriques surface $Y$ we prove that every homology class in $H_1(Y(R), Z/2)$ can be represented by a real algebraic curve if and only if all connected components of $Y(R)$ are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface $Y$.

For manifolds M,M‘ of the form S$^2$ ∪ e$^4$ ∪ e$^6$ we compute the homomorphisms H$_*$M → H$_*$M‘ between homology groups which are realizable by a map F: M → M‘.

Let Φ(x) denote the number of those integers n with φ(n)[les ] x, where φ denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that Φ(x)-Ax [Lt] R(x), where A=ζ(2)ζ(3)/ζ(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell‘s Equation y$^2$=x$^3+$k for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordell‘s Equation in Z for all integers k within the range 0 < | k | [les ] 10 000 and partially extend the computations to 0 < | k | [les ] 100 000. For these values of k, the constant in Hall‘s conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

We develop a ’higher genus‘ analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor $F$ on the category of modular operads, the Feynman transform, which generalizes Kontsevich‘s graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick‘s theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.