Number of rational points of a singular curve

Distribution of Rational Points: A Survey

Rational Points on Singular Intersections of Quadrics

— Given an intersection of two quadrics X ⊂ Pm−1, with m > 9, the quantitative arithmetic of the set X(Q) is investigated under the assumption that the singular locus of X consists of a pair of conjugate singular points defined over Q(i).

Computing singular points of plane rational curves

We compute the singular points of a plane rational curve, parametrically given, using the implicitization matrix derived from the μ-basis of the curve. It is shown that singularity factors, which are defined and uniquely determined by the elementary divisors of the implicitization matrix, contain all the information about the singular points, such as the parameter values of the singular points ...

The Minimal Number of Singular Fibres of Some Fibrations over Rational Curve

The aim of this paper is to study the minimal number of singular fibres of some fibrations. Given a fibration f : S −→ P, where S is a smooth projective surface of general type, it will be proved that f has at least 5 singular fibres and 5 is the best lower bound. If the Kodaira dimension kod(S) = 1, then all fibration f : S −→ P of genus ≥ 2 has at least 4 singular fibres and 4 is also the bes...

Independence of Rational Points on Twists of a given Curve

In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of K-rational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′(K) a...