Related papers: Purely inseparable points on curves

Let $X$ be a semistable curve and $L$ a line bundle whose multidegree is uniform, i.e., in the range between those of the structure sheaf and the dualizing sheaf of $X$. We establish an upper bound for $h^0(X,L)$, which generalizes the…

We give the distribution of points on smooth superelliptic curves over a fixed finite field, as their degree goes to infinity. We also give the distribution of points on smooth m-fold cyclic covers of the line, for any m, as the degree of…

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that for any convex shape $K$, there exist four points on the boundary of $K$ such that the length of any curve…

We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a…

We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and…

We study integral points on affine surfaces by means of a new method, relying on the Subspace Theorem. Under suitable assumptions on the divisor at infinity, we prove that the integral points are contained in a curve. As a corollary, we…

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that…

Let $\mathcal{X}$ be a projective algebraic curve and denote by $\mathcal{X}^{'}$ its strict dual curve. The map $\gamma:\mathcal{X} \longrightarrow \mathcal{X}^{'}$ is called (strict) Gauss map of $\mathcal{X}$. In this manuscript, we…

We bound the j -invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that the modular curve Xsplit (p3) has no non-trivial rational point if p is a sufficiently large…

In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field Fq with many rational points. The results…

We give new arguments that improve the known upper bounds on the maximal number N_q(g) of rational points of a curve of genus g over a finite field F_q for a number of pairs (q,g). Given a pair (q,g) and an integer N, we determine the…

We prove a semistable reduction theorem for principal bundles on curves in almost arbitrary characteristics. For exceptional groups we need some small explicit restrictions on the characteristic.

It has been conjectured that every algebraic curve may be defined either over its field of moduli or over an extension of degree two of it. In this paper we provide a negative answer to it by giving examples of hyperelliptic curves which…

There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for reducible real quintic curves. The classification of real singular points for irreducible real quintic curves is…

In this paper we give a different proof of Kuz'min's result on the number of irreducible polynomials with the first two coefficients fixed. Our technique is to relate the question to the number of points on a curve, and to calculate the…

We provide an explicit bound on the number of periodic points of a rational function defined over a number field, where the bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the…

We give a characterization of nearly free plane curves in terms of their global Tjurina numbers, similar to the characterization of free curves as curves with a maximal Tjurina number, due to A. A. du Plessis and C.T.C. Wall. It is also…

The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied…

In this work we study the properties of maximal and minimal curves of genus 3 over finite fields with discriminant -19. We prove that any such curve can be given by an explicit equation of certain form. Using these equations we obtain a…