Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a complex smooth projective surface and $L$ be a line bundle on $S$. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system $|L|$ with prescribed singularities is a universal polynomial of Chern numbers of $L$ and $S$, assuming $L$ is sufficiently ample. More generally, we show for vector bundles of any rank and smooth varieties of any dimension, similar universal polynomials also exist and equal the number of singular subvarieties cutting out by sections of the vector bundle. This work is a generalization of Göttsche’s conjecture.

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.

This paper is devoted to the calculation of the B-model chiral ring, used in physics, for semiample Calabi–Yau hypersurfaces. We also study the cohomology of semiample hypersurfaces.

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

The Cohen–Lenstra–Martinet heuristics give precise predictions about the class groups of a ’random‘ number field. The 3-rank of quadratic fields is one of the few instances where these have been proven. We prove that, in this case, the rate of convergence is at least sub-exponential. In addition, we show that the defect appearing in Scholz‘s mirror theorem is equidistributed with respect to a twisted Cohen–Lenstra density.

We prove a Reider type theorem for separating any cluster by an adjoint system to a pseudoeffective divisor on a normal surface. As a corollary we get a Reider type theorem for adjoint linear systems (to a nef $\Bbb Q$-divisor) on normal log surfaces. This theorem is new even for smooth surfaces.

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places of $K$. We show that for a given $n\geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate $n$-dimensional quadratic forms over $K$ that have good reduction at all $v\in V$ is finite. This result yields some other finiteness properties, such as the finiteness of the genus $\mathbf{gen}_{K}(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for $i\geqslant 1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type $\mathsf{G}_{2}$.

In this paper, we consider a family of elliptic curves over ${\open Q}$ with 2-torsion part ${\open Z}_2$. We prove that, for every such elliptic curve, a positive proportion of quadratic twists have Mordell–Weil rank 0.

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.

This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier‘s theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal. The local coefficients are related to the Plancherel measures, and their zeroes are shown to be among the poles of the standard intertwining operators.

Let VD be the Shimura curve over $\open Q$ attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of VD and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves over $\bar\open Q$ and over $\open Q$ and we use it to study the set of rational points on VD over quadratic fields. Finally, we obtain explicit equations of elliptic Atkin–Lehner quotients of VD.

This work was largely inspired by a paper of Shustin, in which he proves that for a plane curve of given degree n whose singularities are not too complicated the singularities are versally unfolded by embedding the curve in the space of all curves of degree n; however, our methods are very different. The main result gives fairly explicit lower bounds on the sum of the Tjurina numbers at the singularities of a deformation of a weighted-homogeneous hypersurface, when the deformation is the fibre over an unstable point of an appropriate unfolding. The result is sufficiently flexible to cover a variety of applications, some of which we describe. In particular, we will deduce a generalisation of Shustin‘s result. Properties of discriminant matrices of unfoldings of weighted-homogeneous functions are crucial to the arguments; the parts of the theory needed are described.

Let $G$ be a connected linear algebraic group over a number field $k$. Let $U{\hookrightarrow}X$ be a $G$-equivariant open embedding of a $G$-homogeneous space $U$ with connected stabilizers into a smooth $G$-variety $X$. We prove that $X$ satisfies strong approximation with Brauer–Manin condition off a set $S$ of places of $k$ under either of the following hypotheses:

1. (i) $S$ is the set of archimedean places;

2. (ii) $S$ is a non-empty finite set and $\bar{k}^{\times }=\bar{k}[X]^{\times }$.

The proof builds upon the case $X=U$, which has been the object of several works.

Let x: Ln → $\mathbb S$2n+1 ⊂ $\mathbb R$2n+2 be a minimal submanifold in $\mathbb S$2n+1. In this note, we show that L is Legendrian if and only if for any A ∈ su(n + 1) the restriction to L of 〈Ax, √(−1)x〉 satisfies Δf = 2(n + 1)f. In this case, 2(n + 1) is an eigenvalue of the Laplacian with multiplicity at least ½(n(n + 3)). Moreover if the multiplicity equals to ½(n(n + 3)), then Ln is totally geodesic.

We exhibit a Cremona transformation of $\mathbb{P}^{4}$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.

We prove an explicit formula for the arithmetic intersection number of diagonal cycles on GSpin Rapoport–Zink spaces in the minuscule case. This is a local problem arising from the arithmetic Gan–Gross–Prasad conjecture for orthogonal Shimura varieties. Our formula can be viewed as an orthogonal counterpart of the arithmetic–geometric side of the arithmetic fundamental lemma proved by Rapoport–Terstiege–Zhang in the minuscule case.

Given an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 𝔽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.

For any irreducible quadratic polynomial f(x) in ℤ[x], we obtain the estimate log l.c.m.(f(1),…,f(n))=nlog n+Bn+o(n), where B is a constant depending on f.

We revisit recent work of Heath-Brown on the average order of the quantity r(L1(x))⋯r(L4(x)), for suitable binary linear forms L1,…,L4, as x=(x1,x2) ranges over quite general regions in ℤ2. In addition to improving the error term in Heath-Brown’s estimate, we generalise his result to cover a wider class of linear forms.

We present a simple description of moduli spaces of torsion-free 𝒟-modules (𝒟-bundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with Calogero–Moser quiver varieties. Namely, we show that the moduli of 𝒟-bundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T*X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes T*X[n] in the rank-one case). The proof is based on the description of the derived category of 𝒟-modules on X by a noncommutative version of the Beilinson transform on P1.