Related papers: On a good reduction criterion for proper polycurve…

Let $\Phi$ be an endomorphism of $\SR(\bar{\Q})$, the projective line over the algebraic closure of $\Q$, of degree $\geq2$ defined over a number field $K$. Let $v$ be a non-archimedean valuation of $K$. We say that $\Phi$ has critically…

In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for…

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a…

In this note, we clarify that the boundary criterion for relative cubulation of the first author and Groves works even when the peripheral subgroups are not one-ended. Specifically, if the boundary criterion is satisfied for a relatively…

We consider the question of which quadratic fields have elliptic curves with everywhere good reduction. By revisiting work of Setzer, we expand on congruence conditions that determine the real and imaginary quadratic fields with elliptic…

We bound the length of the periodic part of the orbit of a preperiodic rational subvariety via good reduction information. This bound depends only on the degree of the map, the degree of the subvariety, the dimension of the projective…

In this note, we improve Nikulin's inequality in the case of right-angled hyperbolic polyhedra. The new inequality allows to give much shorter proofs of the known dimension bounds. We also improve Nonaka's lower bound on the number of ideal…

Converting a parametric curve into the implicit form, which is called implicitization, has always been a popular but challenging problem in geometric modeling and related applications. However, the existing methods mostly suffer from the…

In this paper, we prove a `cut-by-curves criterion' for an overconvergent isocrystal on a smooth variety over a field of characteristic $p>0$ to extend logarithmically to its smooth compactification whose complement is a strict normal…

In this paper, we prove the Shafarevich conjecture for proper hyperbolic polycurves, which is a higher dimensional analogue of that for proper hyperbolic curves. First, we study theories of proper hyperbolic polycurves over regular schemes.…

We introduce and motivate a conjecture about the existence of complete, 1-dimensional families of covers of an elliptic curve. If the conjecture holds, then it would imply a uniform lower bound of 5 for slope of the moduli space of curves.…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

This paper describes an efficient approach to constructing a resultant polyline with a minimum number of segments and arcs. While fitting an arc can be done with complexity O(1) (see [1] and [2]), the main complexity is in checking that the…

In this paper, we seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics, i.e. a characterization of curves in positive characteristics which are reduction of Shimura curve over the complex field.…

Let K be a non-archimedean field, and let f in K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sufficient condition, involving only the fixed points of f and their preimages, that determines…

In this paper we study the reduction of Galois covers of curves, from characteristic 0 to characteristic p. The starting point is a is a recent result of Raynaud which gives a criterion for good reduction for covers of the projective line…

Let E be an elliptic curve with additive reduction over the p-adic numbers, and let G be the group of p-adic points on E that have good reduction. This paper gives necessary and sufficient conditions for G to contain non-trivial p-torsion.

In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two…

The purpose of this paper is to give an algorithm for deciding the irreducibility of reduced algebroid curves over an algebraically closed field of arbitrary characteristic. To do this, we introduce a new notion of local tropical variety…

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria…