We study the automorphism groups of the reduction of a modular curve X0(N) over primes p∤N.

A foliation on a non-singular projective variety is algebraically integrable if all leaves are algebraic subvarieties. A non-singular hypersurface X in a non-singular projective variety M equipped with a symplectic form has a naturally defined foliation, called the characteristic foliation on X. We show that if X is of general type and dim M≥4, then the characteristic foliation on X cannot be algebraically integrable. This is a consequence of a more general result on Iitaka dimensions of certain invertible sheaves associated with algebraically integrable foliations by curves. The latter is proved using the positivity of direct image sheaves associated to families of curves.

We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M. A logarithmic resonance variety is also considered and, as an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve. Finally, for a logarithmic 1-form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M. This question was inspired by the recent work by Cohen, Denham, Falk and Varchenko.

If C is a curve of genus 2 defined over a field k and J is its Jacobian, then we can associate a hypersurface K in ℙ3 to J, called the Kummer surface of J. Flynn has made this construction explicit in the case when the characteristic of k is not 2 and C is given by a simplified equation. He has also given explicit versions of several maps defined on the Kummer surface and shown how to perform arithmetic on J using these maps. In this paper we generalize these results to the case of arbitrary characteristic.

The moduli space of n-pointed stable curves of genus g is stratified by the topological type of the curves being parameterized: the closure of the locus of curves with k nodes has codimension k. The one-dimensional components of this stratification are smooth rational curves called F-curves. These are believed to determine all ample divisors. F-conjecture 

A divisor on  is ample if and only if it positively intersects theF-curves.

We study behaviours of the ‘equianharmonic’ parameter of the Grothendieck–Teichmüller group introduced by Lochak and Schneps. Using geometric construction of a certain one-parameter family of quartics, we realize the Galois action on the fundamental group of a punctured Mordell elliptic curve in the standard Galois action on a specific subgroup of the braid group . A consequence is to represent a matrix specialization of the ‘equianharmonic’ parameter in terms of special values of the adelic beta function introduced and studied by Anderson and Ihara.

We introduce the τ-function of a difference rational connection (d-connection) and its isomonodromy transformations. We show that in a continuous limit ourτ-function agrees with the Jimbo–Miwa–Ueno τ-function. We compute the τ-function for the isomonodromy transformations leading to difference Painlevé V and difference Painlevé VI equations. We prove that the gap probability for a wide class of discrete random matrix type models can be viewed as the τ-function for an associated d-connection.

Let k be a field and X a k-form of the projective line. We classify all the isomorphism classes of vector bundles over X.

We show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus (ℂ*)n is a finite group which can be determined using the torsion part of some associated lattice. We also give applications to the study of characteristic varieties of smooth complex algebraic varieties. As an example we discuss A. Suciu’s line arrangement, the so-called deleted B3-arrangement.

We prove that all monomials of κ-classes and ψ-classes are independent in for all . We also give a simple argument for in Rl(ℳg) for .

The enumerative geometry of rth roots of line bundles is crucial in the theory of r-spin curves and occurs in the calculation of Gromov–Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In a previous paper, we constructed a compact moduli stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford’s formula for the Chern character of the Hodge bundle to the direct image of the universal rth root in K-theory.

We study Frobenius groups acting on curves.

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

In this paper we prove that most ropes of arbitrary multiplicity supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1:1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.

We prove the Bertram–Feinberg–Mukai conjecture for a generic curve C of genus g and a semistable vector bundle E of rank two and determinant K on C, namely we prove the injectivity of the Petri-canonical map S2(H0(E))→H0(S2(E)).

We prove that, given a smooth projective curve C of genus g≥2, the forgetful morphism (respectively ) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.

Let X be a smooth projective curve. We consider the dual reductive pair , over X, where H splits on an étale two-sheeted covering . Let BunG (respectively, BunH) be the stack of G-torsors (respectively, H-torsors) on X. We study the functors FG and FH between the derived categories D(BunG) and D(BunH), which are analogs of the classical theta-lifting operators in the framework of the geometric Langlands program. Assume n=m=1 and H nonsplit, that is, with connected. We establish the geometric Langlands functoriality for this pair. Namely, we show that FG :D(BunH)→D(BunG) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups LH→LG. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on BunGL2 and Bessel periods of theta-lifts from to . Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on (one of them makes sense for -modules only).

Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to perform Chabauty- and Brauer–Manin-type calculations for curves of genus 5 with an fixed-point-free involution. As an application, we determine the rational points on a smooth plane quartic and give examples of curves of genus 3 and 5 violating the Hasse principle. The methods are, in principle, applicable to any genus 3 curve with a double cover. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over . By specialization, this also gives examples over .

In this paper, we use the perspective of linear series, and in particular results following from the degeneration tools of limit linear series, to give a number of new results on the existence and non-existence of tamely branched covers of the projective line in positive characteristic. Our results are both in terms of ramification indices and the sharper invariant of monodromy cycles, and the first class of results are obtained by intrinsically algebraic and positive-characteristic arguments.

In this paper is considered the average size of the 2-Selmer groups of a class of quadratic twists of each elliptic curve over ℚ with ℚ-torsion group ℤ2 × ℤ2. The existence is shown of a positive proportion of quadratic twists of such a curve, each of which has rank 0 Mordell-Weil group.