Related papers: Height pairing and nearby cycles
In this short note we give a new upper bound for the size of a set family with a single Hamming distance. Our proof is an application of the linear algebra bound method.
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to the sphere and the hyperbolic…
Let $f : X \to S$ be a smooth projective family defined over $\mathcal{O}_{K}[\mathcal{S}^{-1}]$, where $K \subset \mathbb{C}$ is a number field and $\mathcal{S}$ is a finite set of primes. For each prime $\mathfrak{p} \in…
We consider heights of horizontal irreducible divisors on an arithmetic surface with respect to some hermitian line bundle. We obtain both lower and upper bounds for these heights. The results are different and sometimes stronger that those…
This mostly expository paper records some basic facts about towers of homotopy fiber sequences. We give a proof that a pairing of towers induces a pairing of associated spectral sequences, for towers of spaces and towers of spectra.
We prove a variant of a formula due to S. Zhang relating the Beilinson-Bloch height of the Gross-Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach the height of the Gross-Schoen…
We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of…
We prove that the maximal number of limit cycles which bifurcate from an open period annulus under a given multi-parameter analytic deformation of a given analytic vector field is the same as in an appropriate one-parameter analytic…
We give a criterion for bounding the homological finiteness length of certain HF-groups. This is used in two distinct contexts. Firstly, the homological finiteness length of a non-uniform lattice on a locally finite n-dimensional…
We study the moduli space of pairs $(X,H)$ consisting of a cubic threefold $X$ and a hyperplane $H$ in $\mathbb P^4$. The interest in this moduli comes from two sources: the study of certain weighted hypersurfaces whose middle cohomology…
The Hodge series of a finite matrix group is the generating function for invariant exterior forms of specified order and degree. Lauret, Miatello, and Rossetti gave examples of pairs of non-conjugate cyclic groups having the same Hodge…
The Hodge conjecture is shown to be equivalent to a question about the homology of very ample divisors with ordinary double point singularities. The infinitesimal version of the result is also discussed.
To study problems involving heights as, eg, Manin's conjecture on the number of points of bounded height on an algebraic variety defined over a number field, it is desirable to have a good normalization of these height functions. We show…
We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then…
In this paper, we construct almost resolvable cycle systems of order $4k+1$ for odd $k\ge 11$. This completes the proof of the existence of almost resolvable cycle systems with odd cycle length.
We show that, assuming Vojta's height conjecture, the height of a rational point on an algebraically hyperbolic variety can be bounded "uniformly" in families. This generalizes a result of Su-Ion Ih for curves of genus at least two to…
For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous…
We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat…
It is known that the algebraic \deRham cohomology group $\hDR{i}(X_0/\Q)$ of a nonsingular variety $X_0/\Q$ has the same rank as the rational singular cohomology group $\h^i\sing(\Xh;\Q)$ of the complex manifold $\Xh$ associated to the base…
We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component…