Related papers: Explicit arithmetic intersection theory and comput…

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's…

Given an ordered sequence of $N$-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of $N$ simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the…

Using the connections among almost complete intersection schemes, arithmetically Gorenstein schemes and schemes that are union of complete intersections we give a structure theorem for arithmetically Cohen-Macaulay union of two complete…

We investigate arcs on a pair of pants and present an algorithm to compute the self-intersection number of an arc. Additionally, we establish bounds for the self-intersection number in terms of the word length. We also prove that the…

In this paper, we are concerned with the computations of the $p$-rank of curves in two different setups. We first work with complete intersection varieties in $\mb{P}^n \text{ for}~n\ge 2$ and compute explicitly the action of Frobenius on…

We compute the intersection multiplicities of special cycles in Lubin-Tate spaces, and formulate a new arithmetic fundamental lemma relating these intersections to derivatives of orbital integrals.

Under conditions that prevent tangential intersection, we prove quadratic convergence of a projection algorithm for the feasibility problem of finding a point in the intersection of a smooth curve and line in $\mathbb{R}^2$. This nonconvex…

In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a…

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over Q associated to a…

Any counterexample to the two-dimensional Jacobian Conjecture gives a rational map from one projective plane to another. We use some ideas of the Minimal Model Program to study the combinatorial structure of a rational surface, that is…

This paper proposes new explicit formulas for the doubling and addition step in Miller's algorithm to compute the Tate pairing. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric…

There is a close relationship between the embedded topology of complex plane curves and the (group-theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves which include a special…

We study the correspondence assigning the vertices of a certain quotient of the local Bruhat-Tits tree for the general linear group over a global function field, to conjugacy classes of maximal orders in some quaternion algebras. The…

We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and…

We compute the arithmetic intersection numbers of certain Heegner divisors on integral models of Shimura curves over Q. Our formulas generalize the formulas of Gross-Kohnen-Zagier for intersection numbers of Heegner divisors on integral…

In this paper, we give a Zariski triple of the arrangements for a smooth quartic and its four bitangents. A key criterion to distinguish the topology of such curves is given by a matrix related to the height pairing of rational points…

We give diagrammatic algorithms for computing the group trisection, homology groups, and intersection form of a closed, orientable, smooth 4-manifold, presented as a branched cover of a bridge-trisected surface in $\mathbb{S}^{4}$. The…

This article introduces efficient and user-friendly tools for analyzing the intersection curve between a ringed torus and an irreducible quadric surface. Without loose of generality, it is assumed that the torus is centered at the origin,…

It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the…

We prove an intersection formula for two plane branches in terms of their semigroups and key polynomials. Then we provide a strong version of Bayer's theorem on the set of intersection numbers of two branches and apply it to the logarithmic…