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.

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.

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.

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.