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.

For the special orthogonal group G = SO(2n + 1) over a p-adic field, we consider a discrete series representation of a standard Levi subgroup of G. We prove that the Arthur R-group and the classical R-group of $\pi$ are isomorphic. If $\pi$ is generic, we consider the Aubert involution $\hat{\pi}$. Under the assumption that $\hat{\pi}$ is unitary, we prove that the Arthur R-group of $\hat{\pi}$ is isomorphic to the R-group of $\hat{\pi}$ defined by Ban (Ann. Sci. École Norm. Sup. 35 (2002), 673–693; J. Algebra 271 (2004), 749–767). This is done by establishing the connection between the A-parameters of $\pi$ and $\hat{\pi}$. We prove that the A-parameter of $\hat{\pi}$ is obtained from the A-parameter of $\pi$ by interchanging the two $\textit{SL}(2,\mathbb{C})$ components.

This paper concerns towers of curves over a finite field with many rational points, following Garcia–Stichtenoth and Elkies. We present a new method to produce such towers. A key ingredient is the study of algebraic solutions to Fuchsian differential equations modulo p. We apply our results to towers of modular curves, and find new asymptotically good towers.

for a noetherian scheme, we introduce its unbounded stable derived category. this leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. some applications are included, for instance an analogue of maximal cohen–macaulay approximations, a construction of tate cohomology, and an extension of the classical grothendieck duality. in addition, the relevance of the stable derived category in modular representation theory is indicated.

We prove the unirationality of the moduli space of complex curves of genus 14. The method essentially relies on linkage of curves. In particular it is shown that a general curve of genus 14 admits a projective model D, in a six-dimensional projective space, which is linked to a general curve C of degree 14 and genus 8 by a complete intersection of quadrics. Using this property we are able to obtain, after a reasonable amount of further work, the unirationality result in the case of genus 14. Moreover, some variations of the same method, involving the Hilbert schemes of curves of very low genus, are used to obtain the same result for the known cases of genus 11, 12, 13.

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.

we prove in this paper the green conjecture for generic curves of odd genus. that is, we prove the vanishing $k_{k,1}(x,k_x)=0$ for x a generic curve of genus $2k+1$. this completes our previous work, where the green conjecture for generic curves of genus g with fixed gonality d was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus. the case of generic curves of odd genus was considered as especially important, since hirschowitz and ramanan proved that if the conjecture is true for the generic curve of odd genus, then the locus of jumping syzygies is exactly the locus of exceptional gonality, as predicted by green's conjecture. thus, our result combined with the hirschowitz–ramanan result is a strong confirmation of green's conjecture.

Let Mc be the moduli space of semistable torsion-free sheaves of rank 2 with Chern classes c1 = 0 and c2 = c over a K3 surface with generic polarization. When $c=2n\ge 4$ is even, Mc is a singular projective variety which admits a symplectic form, called the Mukai form, on the smooth part. A natural question raised by O'Grady asks if there exists a desingularization on which the Mukai form extends everywhere nondegenerately. In this paper we show that such a desingularization does not exist for many even integers c by computing the stringy Euler numbers.

We compute the Chow group of 0-cycles on a rational surface defined over a finite extension K of the field $\mathbb{Q}_p$ of p-adic numbers (p a prime) when it is split by an unramified extension of K. We use intersection theory to define a specialisation map so we need to assume that the surface admits a regular proper integral model. A family of examples is worked out to illustrate the method.

We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2-adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2-adic valuation. We formulate an explicit conjecture about what these slopes should be for weight k forms.

given a family x/b of nodal curves we construct canonically and compatibly with base-change, via an explicit blow-up of the cartesian product xr/b, a family wr(x/b) that we show is isomorphic to the relative flag hilbert scheme parametrizing flags of subschemes of fibres of x/b with colengths $1,\dotsc,r$. although wr(x/b) is singular, the important sheaves on it are locally free, which allows us to study some intersection theory on it and deduce enumerative applications, including some relative multiple point formulae enumerating the length-r schemes contained simultaneously in some fibre of x/b and some fibre of a given map from x to a smooth variety.

We prove an interpolation lemma with multiplicities on a commutative algebraic group. This statement is ‘dual’ to zero estimates used in transcendental number theory. The special case where no multiplicities are involved has been proved by Masser.

We compute the mixed Hodge structure on the rational cohomology of the moduli space of smooth genus 4 curves. Specifically, we prove that its Poincaré–Serre polynomial is 1 + t2u2 + t4u4 + t5u6. We show this by producing a stratification of the space, such that all strata are geometric quotients of complements of discriminants.

We prove that, near the boundary of weight space, the 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of ‘evenly spaced’ annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.

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.

The aim of this article is to provide methods for constructing smooth projective complex varieties with ample cotangent bundle. We prove that the intersection of at least n/2 sufficiently ample general hypersurfaces in a complex abelian variety of dimension n has ample cotangent bundle. We also discuss analogous questions for complete intersections in the projective space. Finally, we present an unpublished result of Bogomolov which states that a general linear section of small dimension of a product of sufficiently many smooth projective varieties with big cotangent bundle has ample cotangent bundle.

Let $X\subset \mathbb{P}^r$ be a closed scheme in projective space whose homogeneous ideal is generated by quadrics. We say that X (or its ideal IX) satisfies the condition N2,p if the syzygies of IX are linear for p steps. We show that if X satisfies N2,p then a zero-dimensional or one-dimensional intersection of X with a plane of dimension $\leq p$ is 2-regular. This extends a result of Green and Lazarsfeld. We give conditions when the syzygies of X restrict to the syzygies of the intersection. Many of our results also work for ideals generated by forms of higher degree. As applications, we bound the p for which some well-known projective varieties satisfy N2,p. Another application, carried out by us in a different paper, is a step in the classification of 2-regular reduced projective schemes. Extending a result of Fröberg, we determine which monomial ideals satisfy N2,p. We also apply Green's ‘linear syzygy theorem’ to deduce a relation between the resolutions of IX and $I_{X\cup \Gamma}$ for a scheme Γ, and apply the result to bound the number of intersection points of certain pairs of varieties such as rational normal scrolls.

A differential algebra of finite type over a field $\mathbb{k}$ is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated $\mathbb{k}$-algebra. The prototypical example is the algebra of differential operators on a smooth affine variety, when $\text{char}\mathbb{k} = 0$. We study homological and geometric properties of differential algebras of finite type. The main results concern the rigid dualizing complex over such an algebra A: its existence, structure and variance properties. We also define and study perverse A-modules, and show how they are related to the Auslander property of the rigid dualizing complex of A.

For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduces to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the ‘quotient’ by G reduces a G-CohFT to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the ‘quotient’ by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after rescaling the metric. We then introduce the moduli space of G-stable maps into a smooth, projective variety X with G action. Gromov–Witten-like invariants of these spaces provide the primary source of examples of G-CohFTs. Finally, we explain how these constructions generalize (and unify) the Chen–Ruan orbifold Gromov–Witten invariants of $[X/G]$ as well as the ring $H^{\bullet}(X,G)$ of Fantechi and Göttsche.

We study positivity properties of the moduli (b-)divisor associated to a relative log pair (X, B)/Y with relatively trivial log canonical class.