Related papers: Arithmetic intersection theory over adelic curves
The theory of relative logarithmic jet spaces is developed for log schemes. With this theory the existence of bounds of intersection multiplicities of curves and divisors on certain log schemes is established. This result extends those of…
We apply methods of derived and non-commutative algebraic geometry to understand intersection theoretic phenomena on arithmetic schemes. Specifically, we categorify Bloch's intersection number (in the formulation provided by Kato--Saito).…
We review the following subjects: 1. Basic theory on algebraic curves and their moduli space, 2. Schottky uniformization theory of Riemann surfaces, and its extension called arithmetic uniformization theory, 3. Application to these theories…
We study the class field theory of curve defined over two dimensional local field. The approch used here is a combination of the work of Kato-Saito, and Yoshida where the base field is one dimensional
Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the…
We study the probability for a random line to intersect a given plane curve, defined over a finite field, in a given number of points defined over the same field. In particular, we focus on the limits of these probabilities under successive…
Over the complex numbers, Pl\"ucker's formula computes the number of inflection points of a linear series of projective dimension $r$ and degree $d$ on a curve of genus $g$. Here we explore the geometric meaning of a natural analogue of…
We describe a general algorithm for computing intersection pairings on arithmetic surfaces. We have implemented our algorithm for curves over $\mathbb Q$, and we show how to use it to compute regulators for a number of Jacobians of smooth…
The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$…
Theory for open curves over a local field. After introducing the reciprocity map, we determine the kernel and the cokernel of this map. In addition to this, the Pontrjagin dual of the reciprocity map is also investigated. This gives the one…
An arithmetical structure on a graph is given by a labeling of the vertices which satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical…
In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety. We are going to use their…
Fix an elliptic curve $E_0$ without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of…
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial…
This paper provides a non-standard analogue of Bezout's theorem. This is acheived by showing that, in all characteristics, the notion of Zariski multiplicity coincides with intersection multiplicity when we consider the full families of…
Let R be a commutative, noetherian, local ring. Topological Q-vector spaces modelled on full subcategories of the derived category of R are constructed in order to study intersection multiplicities.
By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a real finite-dimensional linear space. In the paper we give a meticulous and self-contained treatment of this arch-classical subject, which lays a…
We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a "cluster picture" associated to the curve, that describes the p-adic distances between the roots of f(x), and show that…
The arithmetic Chow groups and their product structure are extended from the category of regular arithmetic varieties to regular Deligne-Mumford stacks over the ring of integers in a number field.
The geometric Tevelev degrees of projective space enumerate general, pointed algebraic curves interpolating through the maximal possible number of points. Previous work expresses these invariants in terms of Schubert calculus. Extending…