This paper studies the core of an ideal in a Noetherian local or graded ring. By definition, the core of an ideal is the intersection of all reductions of the ideal. We provide computational formulae for the determination of the core of a graded ring, meaning the core of the unique homogeneous maximal ideal. We then apply the formulae to give answers to several questions raised by Corso, Polini and Ulrich. We are also able to answer in the positive a conjecture raised by these three authors concerning a closed formula for the core. We give a positive answer to their question in the case in which the ring is Cohen–Macaulay with a residue field of characteristic 0, and in the case the ideal is equimultiple.

We study the deformation theory of algebraic stacks. The 2-category of deformations using Ext groups of the cotangent complex is described. Then we show that an algebraic stack has a versal deformation under certain conditions.

This is the second in a sequence of papers on the geometry of spaces of rational curves of degree e on a general hypersurface $X \subset\mathbb{P}^n$ of degree d. In Part I (J. reine angew. Math. 571 (2004), 73–106) it is proved that, if $d<({n+1})/{2}$, then for each e the space of rational curves is irreducible, reduced and has the expected dimension. In this paper it is proved that, if $d^2 + d + 1 \leq n$, then for each e the space of rational curves is a rationally connected variety; in particular it has negative Kodaira dimension.

Let $X\subset {\mathbb P}^n$ be a closed scheme defined by r homogeneous equations of degrees $d_1\ge d_2\ge \dotsb \ge d_r$ over the finite field ${\mathbb F}_q$, with complement $U:={\mathbb P}^n\setminus X$. Let $\kappa$ be the maximum of 0 and the integral part of the rational number $({n-d_2-\dotsb-d_r})/{d_1}$. We show that the eigenvalues of the geometric Frobenius endomorphism acting on the $\ell$-adic cohomology $H^i_{\rm c}(U\times_{{\mathbb F}_q}\overline{{\mathbb F}_q}, {\mathbb Q}_\ell)$ with compact supports are divisible by $q^\kappa$ as algebraic integers.

A p-adically uniformized variety is a smooth projective variety whose associated rigid analytic space admits a uniformization by Drinfeld's p-adic symmetric domain. For such a variety we prove the monodromy-weight conjecture, which asserts that two independently defined filtrations on the log-crystalline cohomology of the special fiber in fact coincide. The proof proceeds by reducing the conjecture to a combinatorial statement about harmonic cochains on the Bruhat–Tits building, which was verified in our previous work.

Let $Q\in{\mathbb C}[x_1,\dotsc,x_n]$ be a homogeneous polynomial of degree $k>0$. We establish a connection between the Bernstein–Sato polynomial $b_Q(s)$ and the degrees of the generators for the top cohomology of the associated Milnor fiber. In particular, the integer $u_Q={\rm max}\{i\in{\mathbb Z}:b_Q(-(i+n)/k)=0\}$ bounds the top degree (as differential form) of the elements in $H^{n-1}_{\rm DR}(Q^{-1}(1),{\mathbb C})$. The link is provided by the relative de Rham complex and ${\mathcal D}$-module algorithms for computing integration functors.

As an application we determine the Bernstein–Sato polynomial $b_Q(s)$ of a generic central arrangement $Q=\prod_{i=1}^kH_i$ of hyperplanes. In turn, we obtain information about the cohomology of the Milnor fiber of such arrangements related to results of Orlik and Randell who investigated the monodromy.

We also introduce certain subschemes of the arrangement determined by the roots of $b_Q(s)$. They appear to correspond to iterated singular loci.

We give a full description of the Chow ring of the complex Cayley plane $\mathbb{O}\mathbb{P}^2$. For this, we describe explicitly the most interesting of its Schubert varieties and compute their intersection products. Translating our results into the Borel presentation, i.e. in terms of Weyl group invariants, we are able to compute the degree of the variety of reductions Y8 introduced by the current authors in arXiv: math.AG/0306328.

We give a purely algebraic proof of the hypersurface case of the Toric Residue Mirror Conjecture recently proposed by Batyrev and Materov.

In 1987, Sabbah proved the existence of Bernstein–Sato polynomials associated with several analytic functions. The purpose of this paper is to give a more elementary and constructive proof of Sabbah's result based on the notion of the analytic Gröbner fan of a $\mathcal D$-module.

This paper shows that a nontrivial uniform decay estimate for complete exponential sums modulo pr, determined by a polynomial map ${\bf P} = (P_1, P_2)$ follows from the existence of a ‘good P decomposition’ of ${\mathbb Z}_p^n$, a property that can be proved with geometric methods, and which was introduced in an earlier article by the present author.

Let X be a complex-projective contact manifold with $b_2(X)=1$. It has long been conjectured that X should then be rational-homogeneous, or equivalently, that there exists an embedding $X \to {\mathbb P}^n$ whose image contains lines.

We show that X is covered by a compact family of rational curves, called ‘contact lines’, that behave very much like the lines on the rational-homogeneous examples: if $x \in X$ is a general point, then all contact lines through x are smooth, no two of them share a common tangent direction at x, and the union of all contact lines through x forms a cone over an irreducible, smooth base. As a corollary, we obtain that the tangent bundle of X is stable.

We show that if $(M,\otimes,I)$ is a monoidal model category then $\mathbb{R}\underline{\rm End}_{M}(I)$ is a (weak) 2-monoid in sSet. This applies in particular when M is the category of A-bimodules over a simplicial monoid A: the derived endomorphisms of A then form its Hochschild cohomology, which therefore becomes a simplicial 2-monoid.

Let P be a principal bundle with semisimple compact simply connected structure group G over a compact simply connected four-manifold M. In this paper we give explicit formulas for the rational homotopy groups and cohomology algebra of the gauge group and of the space of (irreducible) connections modulo gauge transformations for any such bundle.