Related papers: On the arithmetic self-intersection number of the …

In this article we improve the upper bound for the arithmetic self-intersection number of the dualizing sheaf of the minimal regular model for the Fermat curves $F_p$ of prime exponent.

We give an explicitly computable lower bound for the arithmetic self-intersection number of the dualizing sheaf on a large class of arithmetic surfaces. If some technical conditions are satisfied, then this lower bound is positive. In…

In this paper, we introduce numerical cohomology for arithmetic surfaces, which leads to an absolute version of arithmetic Riemann-Roch formula. As an application, we derive an upper bound for the self-intersection number of relative…

We construct the minimal regular model of the Fermat curve of odd squarefree composite exponent $N$ over the $N$-th cyclotomic integers. As an application, we compute upper and lower bounds for the arithmetic self-intersection of the…

We give an explicit formula for the self-intersection number of negative curves on Fermat surfaces. The formula offers us hints to either prove or disprove the Bounded Negativity Conjecture for the Fermat surfaces.

Let K be an algebraic number field and O_K the ring of integers of K. Let f : X --> Spec(O_K) be a stable arithmetic surface over O_K of genus g >= 2. In this short note, we will prove that if f has a reducible geometric fiber, then the…

Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than 4. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number…

Let $N\geq 3$ be a composite, odd, and square-free integer and let $\Gamma$ be the principal congruence subgroup of level $N$. Let $X(N)$ be the modular curve of genus $g_{\Gamma}$ associated to $\Gamma$. In this article, we study the…

We compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve $X_0(p^2)$ over $\mathbb{Q}$. The computation of the self-intersection…

Let $N>1$ be an integer coprime to $6$ such that $N\notin\{5,7,13\}$ and let $g=g(N)$ be the genus of the modular curve $X_0(N)$. We compute the intersection matrices relative to special fibres of the minimal regular model of $X_0(N)$.…

For an oriented surface $S$, the singular set of a fold map $f:S\rightarrow \mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold…

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…

We develop a theory of residues for arithmetic surfaces, establish the reciprocity law around a point, and use the residue maps to explicitly construct the dualizing sheaf of the surface. These are generalisations of known results for…

In this paper, we compute the semi-stable models of modular curves $X_0(p^2)$ for odd primes $p > 3$ and compute the Arakelov self-intersection numbers of the relative dualising sheaves for these models. We give two arithmetic applications…

We compute intersection matrices for modular curves of the form $X_0(p^r)$ with $r \in \{3,4\}$ and as an application, we compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of…

We study lower bounds for the self-intersection of the canonical divisor of "canonical varieties" (i.e. varieties whose canonical linear system gives a birational map). We give some improvements for the known results in the case of surfaces…

We give a close formula for the N\'eron-Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number $\hat{\omega}^2$ of the dualizing sheaf…

This note studies the structure of the divisorial fixed part of the dualizing sheaf of a 1-connected curve D on a smooth surface S. It is shown that if the divisorial fixed part F of the dualizing sheaf is non empty then it has arithmetic…

We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial…

We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact…