Related papers: Explicit arithmetic intersection theory and comput…

Let R be a complete discrete valuation ring with algebraically closed residue field k and fraction field K. Let X_K be a projective smooth and geometrically connected scheme over K. N\'eron defined a canonical pairing on X_K between…

Given a set of planar curves (Jordan arcs), each pair of which meets -- either crosses or touches -- exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has $O(t^2n)$ touchings, where $t$…

We show that the Jacobians of prestable curves over toroidal varieties always admit N\'eron models. These models are rarely quasi-compact or separated, but we also give a complete classification of quasi-compact separated group-models of…

We define an algebraic analogue, in the case of jacobians of curves, of the height jump divisor introduced recently by R. Hain. We give explicit combinatorial formulae for the height jump for families of semistable curves using labelled…

We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to…

A general class of Newton algorithms on Gra{\ss}mann and Lagrange-Gra{\ss}mann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition…

We present algorithms for computing the squared Weil and Tate pairings on elliptic curves and the squared Tate pairing for hyperelliptic curves. The squared pairings introduced in this paper have the advantage that our algorithms for…

For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points. We set this up as a…

We extend previous computations of Calabi-Yau metrics on projective hypersurfaces to free quotients, complete intersections, and free quotients of complete intersections. In particular, we construct these metrics on generic quintics,…

The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…

We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different…

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij--Soederberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences…

We use the invariant theory of binary quartics to give a new formula for the Cassels-Tate pairing on the $2$-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role…

In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using…

We prove a symmetric version of B\'ezout's theorem. More precisely, we show that the symmetric orbit type of a transverse intersection of complex symmetric hypersurfaces in projective space is determined by the degrees. In the projective…

We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.

We compute twists of the modular curve $X(13)$ that parametrise the elliptic curves 13-congruent to a given elliptic curve. Searching for rational points on these twists enables us to find non-trivial pairs of 13-congruent elliptic curves…

Let $X$ be an arithmetic variety over the ring of integers of a number field $K$, with smooth generic fiber $X_K$. We give a formula that relates the dimension of the first Arakelov-Chow vector space of $X$ with the Mordell-Weil rank of the…

Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and…

We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…